Learning Fractional-Order Dynamics from a Single Trajectory

# Learning Fractional-Order Dynamics from a Single Trajectory
Source: [https://arxiv.org/html/2609.18127](https://arxiv.org/html/2609.18127)
Xiaole ZhangAffiliation:Ming Hsieh Department of Electrical and Computer Engineering, University of Southern CaliforniaZiyi ZhangZehao ZhaoAffiliation:Ming Hsieh Department of Electrical and Computer Engineering, University of Southern CaliforniaStephen TuAffiliation:Ming Hsieh Department of Electrical and Computer Engineering, University of Southern CaliforniaGuannan QuAffiliation:Department of Electrical and Computer Engineering, Carnegie Mellon UniversityYorie NakahiraAffiliation:Department of Electrical and Computer Engineering, Carnegie Mellon UniversityPaul BogdanAffiliation:Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California

###### Abstract

Many real\-world processes exhibit long\-range dependence, where the current state depends on a slowly decaying trace of past states rather than on the most recent state alone\. This paper studies system identification for discrete\-time fractional\-order linear time\-invariant systems from a single observed trajectory of lengthtt, a setting that captures such non\-Markovian dynamics through the Grünwald–Letnikov difference operator\. Unlike Markovian systems, fractional\-order systems couple estimation across the entire history, making both statistical analysis and practical identification more challenging\. We propose*Fractional\-Order Ordinary\-Least\-Squares Grid\-Search \(FO\-GS\)*, a simple two\-stage estimator that exploits the diagonal structure of the fractional\-difference operator to decouple the identification problem row\-wise\. Under the stability assumption, we establish high\-probability, non\-asymptotic error bounds for estimating both the fractional order and the system matrix in the heterogeneous setting, with both estimation errors scaling as𝒪\(t−1/2\)\\mathcal\{O\}\(t^\{\-1/2\}\)\. Through experiments, we show that*FO\-GS*outperforms existing baselines in recovering both the fractional order and the underlying system dynamics\.

## 1Introduction

Many complex natural and technological systems exhibit long\-range dependence, a phenomenon in which temporal correlations decay as a power law rather than exponentially over time\. Long\-range dependence has been widely documented across diverse domains, including brain activity\[article,lundstrom2008fractional\], heart\-rate variability\[doi:10\.1073/pnas\.012579499,ivanov1999multifractality\], climate and hydrology\[doi:10\.1061/TACEAT\.0006518,https://doi\.org/10\.1029/WR005i002p00321\], network traffic\[Leland1993OnTS,willinger2003long\], finance\[https://doi\.org/10\.1111/1468\-0262\.00418,DING199383\], and even modern machine\-learning systems such as large language models\[alabdulmohsin2025a,alabdulmohsin2024fractal\]\.

Classical Markovian models are inherently ill\-suited to capture long\-range dependence, as their dynamics depend only on the current state and therefore lack a mechanism to encode persistent historical influence and long\-range correlations\. In contrast, fractional\-order systems provide a natural alternative: the Grünwald–Letnikov difference operator replaces the one\-step recursion from an integer\-order system with a weighted sum over the full history\[hilfer2000applications,ionescu2017role,monje2010fractional,oldham1974fractional\]\. While this nonlocal mathematical formulation makes fractional\-order models particularly well suited for describing history\-dependent dynamics, it also introduces substantial challenges for system identification\. The main challenge in identifying fractional\-order systems is their intrinsic non\-Markovian nature: the current state depends nontrivially on a long history of past states, and the unknown fractional order governs this dependence through the coefficients of the Grünwald–Letnikov difference operator\. Consequently, jointly estimating the fractional order and the system matrix leads to a nonlinear inference problem with long\-range dependence\. Recent works have studied learning and sample\-complexity questions for discrete\-time fractional\-order systems\[chatterjee2022learning,yaghooti2023inferring,11107451,zhang2025endtoend\], but statistical guarantees for learning stochastic fractional\-order systems from a single trajectory remain underexplored\.

![Refer to caption](https://arxiv.org/html/2609.18127v1/NIPS26_sysid_introv3.png)Figure 1:Complex adaptive systems exhibit non\-Markovian dynamics, mathematically characterized by long\-range dependence\. From biological to modern machine learning systems \(left panel\), the autocorrelation function decays as a power law rather than exponentially \(middle panel\)\. Fractional\-order operators with their intrinsic power law memory kernel offer compact mathematical strategies to capture this observed long\-range dependence dynamics \(right panel\)\.To fill this knowledge gap, we study the fractional\-order linear time\-invariant \(FOLTI\) system

Δ𝜶​xs\+1=A​xs\+ηs,\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{s\+1\}=Ax\_\{s\}\+\\eta\_\{s\},whereΔ𝜶\\Delta^\{\\boldsymbol\{\\alpha\}\}denotes the Grünwald–Letnikov difference operator \([1](https://arxiv.org/html/2609.18127#S3.E1)\),𝜶∈\(0,1\)n\\boldsymbol\{\\alpha\}\\in\(0,1\)^\{n\}is the fractional\-order vector,xs∈ℝnx\_\{s\}\\in\\mathbb\{R\}^\{n\}is thenn\-dimensional state at timess, andηs∼𝒩⁡\(0,σ2​In\)\\eta\_\{s\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\_\{n\}\)is the additive Gaussian noise with varianceσ2\\sigma^\{2\}\. We aim to estimate the fractional order𝜶\\boldsymbol\{\\alpha\}and system matrixA∈ℝn×nA\\in\\mathbb\{R\}^\{n\\times n\}given a single observed trajectoryx0,x1,…,xtx\_\{0\},x\_\{1\},\\dots,x\_\{t\}\. We consider both the heterogeneous setting, in which the coordinates may have different fractional order coefficients in the fractional derivatives, and the homogeneous setting, in which all coordinates share a common fractional order\. The central question is whether one can obtain a computationally simple estimator together with non\-asymptotic guarantees in this genuinely non\-Markovian regime\.

This problem has two intertwined challenges: First, the fractional\-order difference operator couples each observation to the entire past trajectory, so standard Markovian identification arguments do not apply directly\. Second, the dependence on the unknown fractional order𝜶\\boldsymbol\{\\alpha\}is nonlinear, making joint optimization over𝜶\\boldsymbol\{\\alpha\}andAAdifficult even in LTI dynamics\. In the single\-trajectory setting, these difficulties are compounded by strong temporal dependence and the absence of independent rollouts\.

Our contributions\.Unlike traditional Ordinary\-Least\-Squares \(OLS\)\-based algorithms for system identification, we propose the*Fractional\-Order OLS Grid\-Search \(FO\-GS\)*algorithm, a two\-stage identification scheme for a stochastic discrete time FOLTI system from a single trajectory\. Using the diagonal structure of the Grünwald–Letnikov difference operator, we identify each row ofAAtogether with each coordinate of𝜶\\boldsymbol\{\\alpha\}separately; along each dimension,*FO\-GS*establishes a grid of all potentialαi\\alpha\_\{i\}with a fixed step sizeϵi\\epsilon\_\{i\}and then estimatesaia\_\{i\}, theii\-th row of system matrixAA, conditioned on each candidateαi\\alpha\_\{i\}\. Then, each\(αi,ai\)\(\\alpha\_\{i\},a\_\{i\}\)\-pair is evaluated to minimize a profiled loss to select the best candidates\. In particular, we show that under the stability assumption,*FO\-GS*provably recovers the ground\-truth parametersα\\alphaandAAwith high probability, with both estimation errors scaling as𝒪\(t−1/2\)\\mathcal\{O\}\(t^\{\-1/2\}\)\.

To the best of our knowledge,*FO\-GS*offers the first non\-asymptotic statistical guarantee for separately identifying𝜶\\boldsymbol\{\\alpha\}andAAfor FOLTI systems on a single trajectory\. We validate*FO\-GS*on both synthetic and real\-world datasets\. Taken together,*FO\-GS*provides a principled framework for learning fractional\-order dynamical systems from limited sequential data, thereby substantially broadening the scope of identifiable system classes beyond the standard Markovian setting\.

## 2Related Work

Our work is rooted in the newly developed branch of fractional\-order system identification, which borrows inspiration from online learning and identification for ordinary LTI systems\.

Learning on a single trajectory for ordinary LTI systems\.The identification of the ordinary LTI systems has a long history of research\[Oymak18,Sarkar18,pmlr\-v75\-simchowitz18a,Sun20,zhang2025learning,zhang2025stabilizing,Zheng201\]\. Compared with those works, our algorithm adapts many methodologies and proof techniques and extends the scope to a broader non\-Markovian system dynamics by coupling traditional OLS\-based identification methods with a grid search scheme on each coordinate of𝜶\\boldsymbol\{\\alpha\}and row ofAA, taking advantage of the diagonal structure of the Grünwald–Letnikov difference operator\. If𝜶=𝟏\\boldsymbol\{\\alpha\}=\\boldsymbol\{1\}and the fractional\-order system simplifies to an ordinary LTI system,*FO\-GS*offers a theoretical guarantee comparable to the state\-of\-the\-art guarantee in ordinary LTI systems\.

Identification of FOLTI systems\.While system identification for ordinary LTI systems is a relatively well understood field, far less attention has been devoted to fractional\-order systems\[1971712334804565506,monje2010fractional,podlubny1998fractional\]\. When multiple trajectories can be sampled, some works have been developed in fractional\-order system identification\.yaghooti2023inferringpropose a two\-stage identification framework that first estimates the fractional\-order parameters from trajectory data generated under a prescribed data\-collection procedure and, conditioned on these estimates, reformulates the discrete\-time control\-affine nonlinear fractional dynamics as a regression problem to infer the unknown system dynamics\.11107451,zhang2025endtoendgeneralize this framework to stochastic settings\.chatterjee2022learningstudy single\-trajectory identification for fractional\-order systems through system truncation and establish a sample\-complexity result for the augmented system matrix\. By contrast, we study the original single\-trajectory identification problem directly, without requiring either data generation or system truncation, and establish the first non\-asymptotic guarantees for both the fractional\-order parameter𝜶\\boldsymbol\{\\alpha\}and the system matrixAA\.

## 3Preliminaries and Problem Formulation

### 3\.1Grünwald–Letnikov Difference Operator

The Grünwald–Letnikov difference operator allows to discretize the fractional\-order derivative and represent it as a finite difference of the form as follows:

Δ𝜶​xs:=∑j=0sΨ⁡\(𝜶,j\)​xs−j,\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{s\}:=\\sum\_\{j=0\}^\{s\}\\Psi\(\\boldsymbol\{\\alpha\},j\)x\_\{s\-j\},\(1\)wherexs∈ℝnx\_\{s\}\\in\\mathbb\{R\}^\{n\},𝜶=\[α1,α2,…,αn\]⊤∈\(0,1\)n\\boldsymbol\{\\alpha\}=\[\\alpha\_\{1\},\\alpha\_\{2\},\\ldots,\\alpha\_\{n\}\]^\{\\top\}\\in\(0,1\)^\{n\}represents the fractional order, andΨ⁡\(𝜶,j\)∈ℝn×n\\Psi\(\\boldsymbol\{\\alpha\},j\)\\in\\mathbb\{R\}^\{n\\times n\}is a diagonal matrix defined asΨ⁡\(𝜶,j\):=diag⁡\(ψ⁡\(α1,j\),ψ⁡\(α2,j\),…,ψ⁡\(αn,j\)\)\\Psi\(\\boldsymbol\{\\alpha\},j\):=\\operatorname\{diag\}\(\\psi\(\\alpha\_\{1\},j\),\\psi\(\\alpha\_\{2\},j\),\\ldots,\\psi\(\\alpha\_\{n\},j\)\)withψ⁡\(αi,j\):=Γ⁡\(j−αi\)Γ⁡\(−αi\)​Γ​\(j\+1\)\\psi\(\\alpha\_\{i\},j\):=\\frac\{\\Gamma\(j\-\\alpha\_\{i\}\)\}\{\\Gamma\(\-\\alpha\_\{i\}\)\\Gamma\(j\+1\)\}fori=1,2,…,n\.i=1,2,\\ldots,n\.HereΓ⁡\(⋅\)\\Gamma\(\\cdot\)denotes the gamma function\.

### 3\.2FOLTI System Identification

The state\-space representation of the discrete\-time FOLTI system reads:

Δ𝜶​xs\+1=A​xs\+ηs,\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{s\+1\}=Ax\_\{s\}\+\\eta\_\{s\},\(2\)wherexs∈ℝnx\_\{s\}\\in\\mathbb\{R\}^\{n\}is the state vector,ηs∼𝒩⁡\(0,σ2​In\)\\eta\_\{s\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\_\{n\}\)for someσ\>0\\sigma\>0is independent and identically distributed, andA∈ℝn×nA\\in\\mathbb\{R\}^\{n\\times n\}is a constant real matrix\. Using the Grünwald–Letnikov difference operator \([1](https://arxiv.org/html/2609.18127#S3.E1)\), we can write the system \([2](https://arxiv.org/html/2609.18127#S3.E2)\) as follows:

xs\+1=A​xs−∑j=1s\+1Ψ⁡\(𝜶,j\)​xs\+1−j\+ηs\.x\_\{s\+1\}=Ax\_\{s\}\-\\sum\_\{j=1\}^\{s\+1\}\\Psi\(\\boldsymbol\{\\alpha\},j\)x\_\{s\+1\-j\}\+\\eta\_\{s\}\.\(3\)The solution to the discrete\-time FOLTI system \([2](https://arxiv.org/html/2609.18127#S3.E2)\) is given by\[guermah2012discrete\]:

xs=Gs​x0\+∑j=0s−1Gs−1−j​ηj,x\_\{s\}=G\_\{s\}x\_\{0\}\+\\sum\_\{j=0\}^\{s\-1\}G\_\{s\-1\-j\}\\eta\_\{j\},where the matricesGsG\_\{s\}are defined recursively by

Gs\\displaystyle G\_\{s\}=\{I,s=0,∑j=0s−1Aj​Gs−1−j,s≥1,Aj=\{A\+diag⁡\(α1,…,αn\),j=0,−Ψ⁡\(𝜶,j\+1\),j≥1\.\\displaystyle=\\begin\{cases\}I,&s=0,\\\\ \\sum\_\{j=0\}^\{s\-1\}A\_\{j\}G\_\{s\-1\-j\},&s\\geq 1,\\end\{cases\}\\qquad A\_\{j\}=\\begin\{cases\}A\+\\operatorname\{diag\}\(\\alpha\_\{1\},\\ldots,\\alpha\_\{n\}\),&j=0,\\\\ \-\\Psi\(\\boldsymbol\{\\alpha\},j\+1\),&j\\geq 1\.\\end\{cases\}\(4\)Problem statement:Given a single observed trajectoryx0,x1,…,xtx\_\{0\},x\_\{1\},\\dots,x\_\{t\}, our goal is to identify the fractional order𝜶\\boldsymbol\{\\alpha\}and the system matrixAA, and to establish statistical guarantees for the resulting estimators\.

## 4Main Results

In this section, we introduce the algorithm for identifying the system parameters𝜶\\boldsymbol\{\\alpha\}andAAin Section[4\.1](https://arxiv.org/html/2609.18127#S4.SS1), and present the sample complexity results for the resulting estimators in Section[4\.2](https://arxiv.org/html/2609.18127#S4.SS2)\.

### 4\.1FO\-GS

Algorithm 1Fractional\-Order OLS Grid\-Search \(FO\-GS\)1:Trajectory

\{xs\}s=0t\\\{x\_\{s\}\\\}\_\{s=0\}^\{t\}, row search interval

\[α¯i,α¯i\]\[\\underline\{\\alpha\}\_\{i\},\\bar\{\\alpha\}\_\{i\}\], grid size

ϵi\\epsilon\_\{i\}\.

2:Estimates

𝜶^=\[α^1,…,α^n\]⊤\\hat\{\\boldsymbol\{\\alpha\}\}=\[\\hat\{\\alpha\}\_\{1\},\\dots,\\hat\{\\alpha\}\_\{n\}\]^\{\\top\}and

A^\\hat\{A\}\.

3:Build grid

𝒜ϵ,i⊂\[α¯i,α¯i\]\\mathcal\{A\}\_\{\\epsilon,i\}\\subset\[\\underline\{\\alpha\}\_\{i\},\\bar\{\\alpha\}\_\{i\}\]\.

4:Form the data matrix

XtX\_\{t\}with \([9](https://arxiv.org/html/2609.18127#S4.E9)\)\.

5:for

i=1,…,ni=1,\\dots,ndo

6:foreach

α∈𝒜ϵ,i\\alpha\\in\\mathcal\{A\}\_\{\\epsilon,i\}do

7:Compute the fractional\-difference row

Δα​Xt\(i\)\\Delta^\{\\alpha\}X\_\{t\}^\{\(i\)\}with \([5](https://arxiv.org/html/2609.18127#S4.E5)\)\.

8:Compute the least\-squares row estimator

a^i​\(α\)\\hat\{a\}\_\{i\}\(\\alpha\)with \([7](https://arxiv.org/html/2609.18127#S4.E7)\)\.

9:Compute the profiled loss

ℒ\(i\)​\(α\)\\mathcal\{L\}^\{\(i\)\}\(\\alpha\)with \([8](https://arxiv.org/html/2609.18127#S4.E8)\)\.

10:endfor

11:Select

α^i←arg⁡minα∈𝒜ϵ,i​ℒ\(i\)​\(α\)\.\\hat\{\\alpha\}\_\{i\}\\leftarrow\\arg\\min\_\{\\alpha\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\mathcal\{L\}^\{\(i\)\}\(\\alpha\)\.
12:Set

a^i←a^i​\(α^i\)\.\\hat\{a\}\_\{i\}\\leftarrow\\hat\{a\}\_\{i\}\(\\hat\{\\alpha\}\_\{i\}\)\.
13:endfor

14:Form

𝜶^←\[α^1,…,α^n\]⊤,\\hat\{\\boldsymbol\{\\alpha\}\}\\leftarrow\[\\hat\{\\alpha\}\_\{1\},\\dots,\\hat\{\\alpha\}\_\{n\}\]^\{\\top\},and

A^←\[a^1⊤,…,a^n⊤\]⊤\.\\hat\{A\}\\leftarrow\[\\hat\{a\}\_\{1\}^\{\\top\},\\dots,\\hat\{a\}\_\{n\}^\{\\top\}\]^\{\\top\}\.return

\(𝜶^,A^\)\(\\hat\{\\boldsymbol\{\\alpha\}\},\\hat\{A\}\)\.

We introduce the algorithm for identifying the system parameters\(𝜶,A\)\(\\boldsymbol\{\\alpha\},A\)of the FOLTI system \([2](https://arxiv.org/html/2609.18127#S3.E2)\) from a single observed trajectory\. The main idea is to isolate each coordinate of𝜶\\boldsymbol\{\\alpha\}under the diagonal structure of the Grünwald–Letnikov difference operator and, for each candidate, solve an OLS problem to estimateAA\. To handle the general fractional\-order setting,[Algorithm1](https://arxiv.org/html/2609.18127#alg1)performs a grid search over each component of𝜶\\boldsymbol\{\\alpha\}, and then solves a row\-wise OLS problem\. Specifically, for eachi∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, define

Δαi​Xt\(i\):=\[Δαi​x1\(i\),…,Δαi​xt\(i\)\]∈ℝ1×t,\\Delta^\{\\alpha\_\{i\}\}X\_\{t\}^\{\(i\)\}:=\\bigl\[\\Delta^\{\\alpha\_\{i\}\}x\_\{1\}^\{\(i\)\},\\dots,\\Delta^\{\\alpha\_\{i\}\}x\_\{t\}^\{\(i\)\}\\bigr\]\\in\\mathbb\{R\}^\{1\\times t\},\(5\)whereΔαi​xs\+1\(i\)=∑j=0s\+1ψ⁡\(αi,j\)​xs\+1−j\(i\)\.\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}=\\sum\_\{j=0\}^\{s\+1\}\\psi\(\\alpha\_\{i\},j\)x\_\{s\+1\-j\}^\{\(i\)\}\.DenoteA=\[a1⊤,…,an⊤\]⊤A=\[a\_\{1\}^\{\\top\},\\dots,a\_\{n\}^\{\\top\}\]^\{\\top\}, whereai∈ℝ1×na\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}represents theii\-th row ofAA\. We further define the loss functionℒ\\mathcal\{L\}as follows:

ℒ⁡\(𝜶,A\)=∑s=0t−1‖Δ𝜶​xs\+1−A​xs‖22=∑i=1n∑s=0t−1\(Δαi​xs\+1\(i\)−ai​xs\)2\.\\mathcal\{L\}\(\\boldsymbol\{\\alpha\},A\)=\\sum\_\{s=0\}^\{t\-1\}\\bigl\\\|\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{s\+1\}\-Ax\_\{s\}\\bigr\\\|\_\{2\}^\{2\}=\\sum\_\{i=1\}^\{n\}\\sum\_\{s=0\}^\{t\-1\}\\bigl\(\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}\\bigr\)^\{2\}\.\(6\)For each rowii, let𝒜i=\[α¯i,α¯i\]⊂\(0,1\]\\mathcal\{A\}\_\{i\}=\[\\underline\{\\alpha\}\_\{i\},\\bar\{\\alpha\}\_\{i\}\]\\subset\(0,1\]be a compact search interval\. Fixing a step sizeϵi\>0\\epsilon\_\{i\}\>0, we construct a uniform grid𝒜ϵ,i=\{αi,k:=α¯i\+k​ϵi\|k∈\{1,…,Mi\}\}⊂𝒜i\\mathcal\{A\}\_\{\\epsilon,i\}=\\bigl\\\{\\alpha\_\{i,k\}:=\\underline\{\\alpha\}\_\{i\}\+k\\epsilon\_\{i\}\\,\|\\,k\\in\\\{1,\\dots,M\_\{i\}\\\}\\bigr\\\}\\subset\\mathcal\{A\}\_\{i\}\. For each candidateαi,k\\alpha\_\{i,k\}, we solve the row\-wise OLS problem and obtain:

a^i​\(αi,k\):=arg⁡minai​ℒ\(i\)​\(αi,k,ai\)=\(Δαi,k​Xt\(i\)\)​Xt⊤​\(Xt​Xt⊤\)−1\.\\hat\{a\}\_\{i\}\(\\alpha\_\{i,k\}\):=\\arg\\min\_\{a\_\{i\}\}\\mathcal\{L\}^\{\(i\)\}\(\\alpha\_\{i,k\},a\_\{i\}\)=\(\\Delta^\{\\alpha\_\{i,k\}\}X\_\{t\}^\{\(i\)\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\.\(7\)We then minimize the corresponding profiled loss and obtain the estimatedαi\\alpha\_\{i\}:

α^i=arg⁡minα∈𝒜ϵ,i​ℒ\(i\)​\(α\)=arg⁡min⁡∑s=0t−1α∈𝒜ϵ,i⁡\(Δα​xs\+1\(i\)−a^i​\(α\)​xs\)2\.\\hat\{\\alpha\}\_\{i\}=\\arg\\min\_\{\\alpha\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\mathcal\{L\}^\{\(i\)\}\(\\alpha\)=\\arg\\min\_\{\\alpha\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\sum\_\{s=0\}^\{t\-1\}\\bigl\(\\Delta^\{\\alpha\}x\_\{s\+1\}^\{\(i\)\}\-\\hat\{a\}\_\{i\}\(\\alpha\)x\_\{s\}\\bigr\)^\{2\}\.\(8\)Finally, by stacking the row estimatorsa^i​\(αi\)\\hat\{a\}\_\{i\}\(\\alpha\_\{i\}\), we obtain the system matrix estimator

A^​\(𝜶\)=\(Δ𝜶​Xt\)​Xt⊤​\(Xt​Xt⊤\)−1,\\hat\{A\}\(\\boldsymbol\{\\alpha\}\)=\(\\Delta^\{\\boldsymbol\{\\alpha\}\}X\_\{t\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\},where

Δ𝜶​Xt=\[Δ𝜶​x1,…,Δ𝜶​xt\],Xt=\[x0,…,xt−1\]\.\\Delta^\{\\boldsymbol\{\\alpha\}\}X\_\{t\}=\[\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{1\},\\ldots,\\Delta^\{\\boldsymbol\{\\alpha\}\}x\_\{t\}\],\\;X\_\{t\}=\[x\_\{0\},\\ldots,x\_\{t\-1\}\]\.\(9\)The commensurate setting is a direct specialization of the above procedure\. When all coordinates share a common fractional order, i\.e\.,𝜶=α​𝟏\\boldsymbol\{\\alpha\}=\\alpha\\boldsymbol\{1\}, the row\-wise searches collapse to a single one\-dimensional search overα\\alpha\.

### 4\.2Theoretical Guarantees

In this section, we present complexity guarantees for the proposed algorithm\. We begin by introducing the assumptions required for our main results\. For the rest of the paper, we use𝜶⋆=\[α1,⋆,…​αn,⋆\]⊤\\boldsymbol\{\\alpha\}\_\{\\star\}=\[\\alpha\_\{1,\\star\},\.\.\.\\alpha\_\{n,\\star\}\]^\{\\top\}to denote the true fractional\-order vector andA⋆A\_\{\\star\}to denote the true system matrix\. We assumex0=0x\_\{0\}=0for simplicity\. We first introduce a stability assumption that is standard in the LTI system identification literature\.

###### Assumption 1\.

The system parameters\(A⋆,𝜶⋆\)\(A\_\{\\star\},\\boldsymbol\{\\alpha\}\_\{\\star\}\)are stable in the sense that for all\|z\|≤1\|z\|\\leq 1,

det\(diag⁡\(\(1−z\)α1,⋆,…,\(1−z\)αn,⋆\)−z​A⋆\)≠0\.\\det\\\!\\Bigl\(\\operatorname\{diag\}\\bigl\(\(1\-z\)^\{\\alpha\_\{1,\\star\}\},\\dots,\(1\-z\)^\{\\alpha\_\{n,\\star\}\}\\bigr\)\-zA\_\{\\star\}\\Bigr\)\\neq 0\.

[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)can be interpreted as the fractional\-order version of the stability assumption common in control literature\[jedra2020finitetimeidentificationstablelinear,oymak2019nonasymptoticidentificationltisystems,petravs2021stability,rivero2013stability,sarkar2021finite\]\. It is slightly stronger than minimal stability, since imposing the condition atz=1z=1implies thatA⋆A\_\{\\star\}is invertible\. This property is used in our analysis to bound the matricesGsG\_\{s\}in \([4](https://arxiv.org/html/2609.18127#S3.E4)\)\. It ensures that the statextx\_\{t\}does not blow\-up with time\. If the system is unstable, then any error at the early time period would be exponentially amplified by the unstable system dynamics in a phenomenon known as*stochastic coupling*\[zhang2025learning\]\. We leave this as a future direction of this paper\.

We are now ready to introduce the main theorems\. First, we introduce the error bound on𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}:

###### Theorem 1\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds and the population separation gapγ\\gammain \([64](https://arxiv.org/html/2609.18127#A2.E64)\) is positive\. Letϵmax:=max1≤i≤n⁡ϵi\.\\epsilon\_\{\\max\}:=\\max\_\{1\\leq i\\leq n\}\\epsilon\_\{i\}\.Fixδ∈\(0,1/2\)\\delta\\in\(0,1/2\)\. Ift≳1mini⁡α¯i4​\(n\+log⁡∑i=1nMiδ\),t\\gtrsim\\frac\{1\}\{\\min\_\{i\}\\underline\{\\alpha\}\_\{i\}^\{4\}\}\\left\(n\+\\log\\frac\{\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\\right\),and the excitation \([65](https://arxiv.org/html/2609.18127#A2.E65)\) and localization \([66](https://arxiv.org/html/2609.18127#A2.E66)\) conditions are satisfied, then with probability at least1−δ1\-\\delta,

‖𝜶^−𝜶⋆‖∞2≲poly⁡\(n,1δ\)​\[ϵmax2\+1t​∑i=1nlog⁡Mi​nδ\+1t​log⁡nδ\],\\left\\\|\\widehat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\lesssim\\operatorname\{poly\}\\\!\\left\(n,\\frac\{1\}\{\\delta\}\\right\)\\left\[\\epsilon\_\{\\max\}^\{2\}\+\\frac\{1\}\{t\}\\sum\_\{i=1\}^\{n\}\\log\\frac\{M\_\{i\}n\}\{\\delta\}\+\\frac\{1\}\{t\}\\log\\frac\{n\}\{\\delta\}\\right\],\(10\)where≲\\lesssimhides system\-dependent constants independent oftt,nn,δ\\delta,ϵmax\\epsilon\_\{\\max\}, andMiM\_\{i\}\. Consequently, ifϵmax=𝒪\(t−1/2\),\\epsilon\_\{\\max\}=\\mathcal\{O\}\\left\(t^\{\-1/2\}\\right\),then‖𝛂^−𝛂⋆‖∞=𝒪\(t−1/2\)\.\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}=\\mathcal\{O\}\\left\(t^\{\-1/2\}\\right\)\.

We defer the proof of[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)to[AppendixB](https://arxiv.org/html/2609.18127#A2)\. The condition forttrequires the trajectory to be long enough for both the global and local lower isometry bounds to hold uniformly over the finite search grid\. In addition to the excitation and localization conditions, the trajectory length scales ast≳1mini⁡α¯i4​\(n\+log⁡∑i=1nMiδ\),t\\gtrsim\\frac\{1\}\{\\min\_\{i\}\\underline\{\\alpha\}\_\{i\}^\{4\}\}\\left\(n\+\\log\\frac\{\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\\right\),thus smaller fractional orders require longer trajectories, reflecting the stronger long\-memory dependence in this regime\.[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)makes explicit the tradeoff between statistical error and grid discretization error\. Up to logarithmic factors in the grid sizeMiM\_\{i\}, the statistical term decays ast−1/2t^\{\-1/2\}, whereas the discretization term decays asϵmax\\epsilon\_\{\\max\}\. Accordingly, choosingϵmax=𝒪\(t−1/2\)\\epsilon\_\{\\max\}=\\mathcal\{O\}\(t^\{\-1/2\}\)makes the two contributions comparable\. For a uniform grid over a bounded interval, this corresponds toMi=𝒪⁡\(t1/2\)M\_\{i\}=\\mathcal\{O\}\(t^\{1/2\}\), which is sufficient to match the statistical precision\. To the best of our knowledge, this is the first high\-probability single\-trajectory error bound for estimation of the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}in the FOLTI setting\. Our result is complementary to prior work based on truncated, bisection\-like identification schemes\[chatterjee2022learning\], and to more recent analyses developed under different data\-generation frameworks\[yaghooti2023inferring,11107451,zhang2025endtoend\]\. We then discuss the error complexity of estimatingA⋆A\_\{\\star\}in the following theorem:

###### Theorem 2\.

Under the same condition as in[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1), fixδ∈\(0,12\)\\delta\\in\(0,\\frac\{1\}\{2\}\)and consider the system \([3](https://arxiv.org/html/2609.18127#S3.E3)\)\. LetΓs=∑m=0s−1Gm​Gm⊤\\Gamma\_\{s\}=\\sum\_\{m=0\}^\{s\-1\}G\_\{m\}G\_\{m\}^\{\\top\}andΞt\(δ,k\):=nlog9​nδ\+logdet\(ΓtΓk−1\)\.\\Xi\_\{t\}\(\\delta,k\):=n\\log\\frac\{9n\}\{\\delta\}\+\\log\\det\\\!\\left\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\\right\)\.Then there exist universal constantsc,C\>0c,C\>0such that, for any integerkksatisfyingtk≥c​Ξt​\(δ,k\),\\frac\{t\}\{k\}\\geq c\\Xi\_\{t\}\(\\delta,k\),the following holds with probability at least1−δ1\-\\delta:

‖A^​\(𝜶^\)−A⋆‖op≤C⁡\(Ξt​\(δ,k\)t​λmin​\(Γk\)\+S1​‖𝜶^−𝜶⋆‖∞​n​C~G2δ​λmin​\(Γ⌊k/2⌋\)\),\\displaystyle\\bigl\\\|\\hat\{A\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\-A\_\{\\star\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\leq C\\Biggl\(\\sqrt\{\\frac\{\\Xi\_\{t\}\(\\delta,k\)\}\{t\\,\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\}\+S\_\{1\}\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\sqrt\{\\frac\{n\\tilde\{C\}\_\{G\}^\{2\}\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{\{\\lfloor k/2\\rfloor\}\}\)\}\}\\Biggr\),whereC~G\\tilde\{C\}\_\{G\}is a constant depending on\(A⋆,𝛂⋆\)\(A\_\{\\star\},\\boldsymbol\{\\alpha\}\_\{\\star\}\), andS1S\_\{1\}is a constant depending on the grid search interval\.

We defer the proof of[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2)to[AppendixC](https://arxiv.org/html/2609.18127#A3)\.[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2)shows that the estimation error forA⋆A\_\{\\star\}decomposes into two parts\. The first term is a standard oracle least\-squares error, which is the error that would arise even if the true fractional\-order vector were known\. The second term quantifies the propagation of the fractional\-order estimation error into the estimation ofA⋆A\_\{\\star\}\. Consequently, when𝜶^\\hat\{\\boldsymbol\{\\alpha\}\}is obtained from[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1), the bound forA⋆A\_\{\\star\}inherits the same statistical\-discretization tradeoff as the bound for𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}\. In particular, ifϵmax=𝒪\(t−1/2\)\\epsilon\_\{\\max\}=\\mathcal\{O\}\(t^\{\-1/2\}\), then the propagated term is of ordert−1/2t^\{\-1/2\}, while the oracle term is also of ordert−1/2t^\{\-1/2\}\. Therefore, the overall estimation error forA⋆A\_\{\\star\}achieves thet−1/2t^\{\-1/2\}rate\.chatterjee2022learninganalyzes a truncated system identification scheme, but does not explicitly quantify the estimation error for the original system matrixA⋆A\_\{\\star\}or how fractional\-order estimation error propagates to the estimation ofA⋆A\_\{\\star\}\. We also note that in the case when𝜶⋆=𝟏\\boldsymbol\{\\alpha\}\_\{\\star\}=\\mathbf\{1\}is known a priori, there is no need to estimate the fractional\-order parameter\. So[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2)is equivalent to the current state\-of\-the\-art bound for estimatingA⋆A\_\{\\star\}for ordinary LTI systems\[pmlr\-v75\-simchowitz18a\]\.

## 5Proof Outline

The proof in this paper is split into two steps, bounding the estimation error for𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}and then forA⋆A\_\{\\star\}\. In this section, we provide an outline of the proof for each, and defer the details to the appendix\.

### 5\.1Proof of[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1): bounding the error of𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}

The estimation of𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}can be analyzed in two steps: controlling the in\-sample error and establishing a lower isometry bound\. The first step gives an absorbable upper bound on the in\-sample prediction error, while the second localizes the estimator and converts the same error into a quadratic lower bound on the fractional\-order estimation error\.

Controlling the in\-sample error\.For each rowii, definebs\(i\)​\(αi\):=\(Δαi−Δαi,⋆\)​xs\+1\(i\)b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\left\(\\Delta^\{\\alpha\_\{i\}\}\-\\Delta^\{\\alpha\_\{i,\\star\}\}\\right\)x\_\{s\+1\}^\{\(i\)\}andys\(i\)​\(αi\):=bs\(i\)​\(αi\)\+2​ηs\(i\)y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\+2\\eta\_\{s\}^\{\(i\)\}, and letαi∘\\alpha\_\{i\}^\{\\circ\}denote the grid point closest toαi,⋆\\alpha\_\{i,\\star\}\. We study the unnormalized in\-sample error

ℰt:=∑s=0t−1‖\(Δ𝜶^−Δ𝜶⋆\)​xs\+1−\(A^−A⋆\)​xs‖22\.\\mathcal\{E\}\_\{t\}:=\\sum\_\{s=0\}^\{t\-1\}\\left\\\|\\left\(\\Delta^\{\\hat\{\\boldsymbol\{\\alpha\}\}\}\-\\Delta^\{\\boldsymbol\{\\alpha\}\_\{\\star\}\}\\right\)x\_\{s\+1\}\-\(\\hat\{A\}\-A\_\{\\star\}\)x\_\{s\}\\right\\\|\_\{2\}^\{2\}\.By the optimality of the row\-wise estimatorL\(i\)​\(α^i,a^i\)≤L\(i\)​\(αi∘,a^i​\(αi∘\)\)≤L\(i\)​\(αi∘,ai,⋆\)L^\{\(i\)\}\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\)\\leq L^\{\(i\)\}\\\!\\left\(\\alpha\_\{i\}^\{\\circ\},\\hat\{a\}\_\{i\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\)\\leq L^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\},a\_\{i,\\star\}\)and a quadratic maximization over the system matrix perturbation, we obtain

ℰt≤∑i=1nmaxαi∈𝒜ϵ,i\\displaystyle\\mathcal\{E\}\_\{t\}\\leq\\sum\_\{i=1\}^\{n\}\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\{−4∑s=0t−1ηs\(i\)bs\(i\)\(αi\)−∑s=0t−1\|bs\(i\)\(αi\)\|2⏟Ut,i​\(αi\)\+‖\(∑s=0t−1ys\(i\)​\(αi\)​xs⊤\)​\(Xt​Xt⊤\)−12‖22⏟Vt,i​\(αi\)\}\\displaystyle\\left\\\{\\underbrace\{\-4\\sum\_\{s=0\}^\{t\-1\}\\eta\_\{s\}^\{\(i\)\}b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\sum\_\{s=0\}^\{t\-1\}\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\}\_\{U\_\{t,i\}\(\\alpha\_\{i\}\)\}\+\\underbrace\{\\left\\\|\\left\(\\sum\_\{s=0\}^\{t\-1\}y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)x\_\{s\}^\{\\top\}\\right\)\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-\\frac\{1\}\{2\}\}\\right\\\|\_\{2\}^\{2\}\}\_\{V\_\{t,i\}\(\\alpha\_\{i\}\)\}\\right\\\}\+∑i=1n\[4​∑s=0t−1ηs\(i\)​bs\(i\)​\(αi∘\)\+2​∑s=0t−1\|bs\(i\)​\(αi∘\)\|2\]⏟Γtgrid\.\\displaystyle\+\\underbrace\{\\sum\_\{i=1\}^\{n\}\\left\[4\\sum\_\{s=0\}^\{t\-1\}\\eta\_\{s\}^\{\(i\)\}b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\+2\\sum\_\{s=0\}^\{t\-1\}\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\|^\{2\}\\right\]\}\_\{\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}\}\.\(11\)
Exact\-grid case\.If the true parameter lies exactly on the search grid, i\.e\.,αi,⋆∈𝒜ϵ,i\\alpha\_\{i,\\star\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}for everyii, then we may takeαi∘=αi,⋆\\alpha\_\{i\}^\{\\circ\}=\\alpha\_\{i,\\star\}\. In this casebs\(i\)​\(αi∘\)=0b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)=0for everyiiandss, and henceΓtgrid=0\.\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}=0\.Thus, the continuous empirical risk minimizer offset inequality is recovered as a special case\.

Row\-wise complexity decomposition\.Since the fractional\-order operator is diagonal across state coordinates and the estimator searches for eachαi\\alpha\_\{i\}separately, the offset complexity decomposes as

∑i=1nmaxαi∈𝒜ϵ,i⁡\{Ut,i​\(αi\)\+Vt,i​\(αi\)\},\\sum\_\{i=1\}^\{n\}\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\left\\\{U\_\{t,i\}\(\\alpha\_\{i\}\)\+V\_\{t,i\}\(\\alpha\_\{i\}\)\\right\\\},rather than requiring a supremum over the full Cartesian grid𝒜ϵ,1×⋯×𝒜ϵ,n\\mathcal\{A\}\_\{\\epsilon,1\}\\times\\cdots\\times\\mathcal\{A\}\_\{\\epsilon,n\}\. This row\-wise decomposition is important for obtaining the sharp complexity dependence of the grid\-search estimator\.

The offset martingale complexity argument\[pmlr\-v178\-ziemann22a\]preserves the negative quadratic term inUt,iU\_\{t,i\}, yielding a term proportional to‖𝜶^−𝜶⋆‖∞2\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}with a tunable coefficient, which can later be absorbed by the lower isometry bound\. The additional termΓtgrid\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}accounts for the finite\-grid approximation and vanishes when the true fractional orders lie on the search grid\. BoundingUt,iU\_\{t,i\},Vt,iV\_\{t,i\}, andΓtgrid\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}separately then yields the following high\-probability control ofℰt\\mathcal\{E\}\_\{t\}\.

###### Lemma 1\.

For any fixedτ∈\(0,1\]\\tau\\in\(0,1\]andρ\>0\\rho\>0, ift/k≥c​Ξt​\(δ,k\)t/k\\geq c\\Xi\_\{t\}\(\\delta,k\), then with probability at least1−δ1\-\\delta,

ℰt≤\\displaystyle\\mathcal\{E\}\_\{t\}\\leq\{\}\(τ\+ρ\)​S12​‖𝜶^−𝜶⋆‖∞2​tr⁡\(Xt​Xt⊤\)\+3​S12​ϵmax2​tr⁡\(Xt​Xt⊤\)\\displaystyle\(\\tau\+\\rho\)S\_\{1\}^\{2\}\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\+3S\_\{1\}^\{2\}\\epsilon\_\{\\max\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\+4​\(1\+ρ−1\)​C2​tr⁡\(Xt​Xt⊤\)t​λmin​\(Γk\)​Ξt​\(δ,k\)\+8​σ2τ​∑ilog⁡3​Mi​nδ\+8​σ2​log⁡3δ\.\\displaystyle\+\\frac\{4\(1\+\\rho^\{\-1\}\)C^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\{t\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\)\+\\frac\{8\\sigma^\{2\}\}\{\\tau\}\\sum\_\{i\}\\log\\frac\{3M\_\{i\}n\}\{\\delta\}\+8\\sigma^\{2\}\\log\\frac\{3\}\{\\delta\}\.The coefficientτ\+ρ\\tau\+\\rhois tunable and can be absorbed by lower isometry\.

We defer the proof of[Lemma1](https://arxiv.org/html/2609.18127#Thmlemma1)to[SectionB\.1](https://arxiv.org/html/2609.18127#A2.SS1)\.

Lower isometry\.For each rowii, define the unprofiled and profiled noiseless errors𝒬t,i​\(αi,v\):=∑s=0t−1\|bs\(i\)​\(αi\)−v​xs\|2\\mathcal\{Q\}\_\{t,i\}\(\\alpha\_\{i\},v\):=\\sum\_\{s=0\}^\{t\-1\}\\bigl\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-vx\_\{s\}\\bigr\|^\{2\}and𝒬¯t,i​\(αi\):=infv∈ℝ1×n𝒬t,i​\(αi,v\),\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\):=\\inf\_\{v\\in\\mathbb\{R\}^\{1\\times n\}\}\\mathcal\{Q\}\_\{t,i\}\(\\alpha\_\{i\},v\),and the row\-wise population riskRt\(i\)​\(αi\):=1t​infai∈ℝ1×n∑s=0t−1𝔼⁡\[\(Δαi​xs\+1\(i\)−ai​xs\)2\]\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\frac\{1\}\{t\}\\inf\_\{a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\\left\(\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}\\right\)^\{2\}\\right\]\.A global profiled lower isometry bound shows that, uniformly over grid points outside the local neighborhood ofαi,⋆,\\alpha\_\{i,\\star\},

𝒬¯t,i​\(αi\)≥t2​\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\}\{2\}\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\.Together with the upper bound onℰt\\mathcal\{E\}\_\{t\}in[Lemma1](https://arxiv.org/html/2609.18127#Thmlemma1), this excludes grid points outside the local neighborhood and localizesα^i\\hat\{\\alpha\}\_\{i\}to a set𝒢i\\mathcal\{G\}\_\{i\}in \([34](https://arxiv.org/html/2609.18127#A2.E34)\) aroundαi,⋆\\alpha\_\{i,\\star\}, where the first\-order expansion ofbs\(i\)​\(αi\)b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)has controlled remainder and the corresponding population derivative curvatureμt,i\\mu\_\{t,i\}is nondegenerate\. Within𝒢i\\mathcal\{G\}\_\{i\}, this yields the quadratic lower\-isometry bound𝒬¯t,i​\(αi\)≥t​μt,i8​\|αi−αi,⋆\|2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.The following lemma formalizes this local lower\-isometry property\.

###### Lemma 2\.

Under[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1), ift≳1mini⁡α¯i4​\(n\+log⁡∑i=1nMiδ\),t\\gtrsim\\frac\{1\}\{\\min\_\{i\}\\underline\{\\alpha\}\_\{i\}^\{4\}\}\\left\(n\+\\log\\frac\{\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\\right\),then with probability at least1−δ1\-\\delta, simultaneously for alli∈\[n\]i\\in\[n\]and allαi∈𝒢i\\alpha\_\{i\}\\in\\mathcal\{G\}\_\{i\},𝒬¯t,i​\(αi\)≥t​μt,i8​\|αi−αi,⋆\|2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.Consequently, on any event for whichα^i∈𝒢i\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\}for everyi∈\[n\]i\\in\[n\],𝒬t,i​\(α^i,a^i−ai,⋆\)≥t​μt,i8​\|α^i−αi,⋆\|2\.\\mathcal\{Q\}\_\{t,i\}\\left\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\right\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.

We defer the proof of[Lemma2](https://arxiv.org/html/2609.18127#Thmlemma2)to[SectionB\.2](https://arxiv.org/html/2609.18127#A2.SS2)\. Combining the lower isometry bound in[Lemma2](https://arxiv.org/html/2609.18127#Thmlemma2)with the in\-sample upper bound in[Lemma1](https://arxiv.org/html/2609.18127#Thmlemma1)and choosing the tunable coefficient sufficiently small gives

‖𝜶^−𝜶⋆‖∞2≲ϵmax2\+𝒪⁡\(t−1\)\.\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\lesssim\\epsilon\_\{\\max\}^\{2\}\+\\mathcal\{O\}\(t^\{\-1\}\)\.Hence, choosingϵmax=O\(t−1/2\)\\epsilon\_\{\\max\}=O\(t^\{\-1/2\}\)yields thet−1/2t^\{\-1/2\}rate in[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)\.

### 5\.2Proof of[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2): bounding the error ofA⋆A\_\{\\star\}

We decompose the identification error ofAAgiven𝜶\\boldsymbol\{\\alpha\}\. For any𝜶\\boldsymbol\{\\alpha\}, we have

A^​\(𝜶\)−A∗\\displaystyle\\hat\{A\}\(\\boldsymbol\{\\alpha\}\)\-A\_\{\*\}=\(Bt​\(𝜶\)\+Wt\)​Xt⊤​\(Xt​Xt⊤\)−1,\\displaystyle=\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+W\_\{t\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\},whereWt:=\[η0,…,ηt−1\]∈ℝn×tW\_\{t\}:=\[\\eta\_\{0\},\\dots,\\eta\_\{t\-1\}\]\\in\\mathbb\{R\}^\{n\\times t\},Bt​\(𝜶\):=\[b0​\(𝜶\),…,bt−1​\(𝜶\)\]∈ℝn×t,B\_\{t\}\(\\boldsymbol\{\\alpha\}\):=\[\\,b\_\{0\}\(\\boldsymbol\{\\alpha\}\),\\ldots,b\_\{t\-1\}\(\\boldsymbol\{\\alpha\}\)\\,\]\\in\\mathbb\{R\}^\{n\\times t\},andbs​\(𝜶\)=∑j=0s\(Ψ⁡\(𝜶,j\)−Ψ⁡\(𝜶⋆,j\)\)​xs−j=\[bs\(1\)​\(𝜶\),…,bs\(n\)​\(𝜶\)\]⊤\.b\_\{s\}\(\\boldsymbol\{\\alpha\}\)=\\sum\_\{j=0\}^\{s\}\\bigl\(\\Psi\(\\boldsymbol\{\\alpha\},j\)\-\\Psi\(\\boldsymbol\{\\alpha\}\_\{\\star\},j\)\\bigr\)\\,x\_\{s\-j\}=\[b\_\{s\}^\{\(1\)\}\(\\boldsymbol\{\\alpha\}\),\\ldots,b\_\{s\}^\{\(n\)\}\(\\boldsymbol\{\\alpha\}\)\]^\{\\top\}\.Thus

‖A^​\(𝜶\)−A⋆‖op≤‖Wt​Xt⊤​\(Xt​Xt⊤\)−1‖op⏟noise term\+‖Bt​\(𝜶\)​Xt⊤​\(Xt​Xt⊤\)−1‖op⏟bias term\.\\bigl\\\|\\hat\{A\}\(\\boldsymbol\{\\alpha\}\)\-A\_\{\\star\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\;\\leq\\;\\underbrace\{\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\}\_\{\\text\{noise term\}\}\\;\+\\;\\underbrace\{\\bigl\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\}\_\{\\text\{bias term\}\}\.\(12\)We therefore decompose the identification error in \([12](https://arxiv.org/html/2609.18127#S5.E12)\) into noise and bias terms, which we bound separately in[Lemmas3](https://arxiv.org/html/2609.18127#Thmlemma3)and[4](https://arxiv.org/html/2609.18127#Thmlemma4)\. To bound the noise term, we adapt the technique fromSarkar18,pmlr\-v75\-simchowitz18aand get the following lemma:

###### Lemma 3\.

Fixδ∈\(0,12\)\\delta\\in\(0,\\frac\{1\}\{2\}\)and consider the system \([3](https://arxiv.org/html/2609.18127#S3.E3)\) under[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)\. Then there exist universal constantsc,C\>0c,C\>0such that

ℙ\[∥WtXt⊤\(XtXt⊤\)−1∥op\>Ct​λmin​\(Γk\)nlognδ\+logdet\(ΓtΓk−1\)\]≤δ,\\mathbb\{P\}\\\!\\left\[\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\>\\frac\{C\}\{\\sqrt\{t\\,\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\}\\sqrt\{n\\log\\frac\{n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\}\\right\]\\leq\\delta,for anykksuch thattk≥c\(nlog\(n/δ\)\+logdet\(ΓtΓk−1\)\)\\frac\{t\}\{k\}\\geq c\\\!\\left\(n\\log\(n/\\delta\)\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)holds\.

We defer the proof of[Lemma3](https://arxiv.org/html/2609.18127#Thmlemma3)to[SectionC\.1](https://arxiv.org/html/2609.18127#A3.SS1)\. We now offer the bound for the bias term in \([12](https://arxiv.org/html/2609.18127#S5.E12)\):

###### Lemma 4\.

With probability at least1−2​δ1\-2\\delta, the following holds

‖Bt​\(𝜶\)​Xt⊤​\(Xt​Xt⊤\)−1‖op≤S1​‖𝜶−𝜶⋆‖∞​320​n​C~G29​δ​p2​λmin​\(Γ⌊k/2⌋\),\\bigl\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\leq S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\sqrt\{\\frac\{320n\\tilde\{C\}\_\{G\}^\{2\}\}\{9\\delta\{p^\{2\}\}\\lambda\_\{\\min\}\(\\Gamma\_\{\{\\lfloor k/2\\rfloor\}\}\)\}\},whereαmin=min1≤i≤n⁡αi,⋆\\alpha\_\{\\min\}=\\min\_\{1\\leq i\\leq n\}\\alpha\_\{i,\\star\}, andp=320p=\\frac\{3\}\{20\}\.

We defer the proof of[Lemma4](https://arxiv.org/html/2609.18127#Thmlemma4)to[SectionC\.2](https://arxiv.org/html/2609.18127#A3.SS2)\.

## 6Experiments

We evaluate the proposed method*FO\-GS*through two sets of experiments\. First, we use synthetic data to validate the theoretical guarantee and compare*FO\-GS*with existing fractional\-order identification algorithms\[chatterjee2022learning,flandrin2002wavelet\]\.*FO\-WT*estimates the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}using a wavelet\-based technique\[flandrin2002wavelet\]and then applies OLS to estimate the system matrixA⋆A\_\{\\star\}\.*FO\-BS*estimates𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}via binary search and then identifiesA⋆A\_\{\\star\}by applying OLS to an augmented\-state representation obtained through system truncation\[chatterjee2022learning\]\. Second, as fractional\-order systems have been used in analyzing electroencephalogram \(EEG\) data, we further demonstrate that*FO\-GS*excels in minimizing the one\-step normalized mean squared error \(NMSE\) on both training and testing datasets in comparison to both existing methods \(*FO\-BS*and*FO\-WT*\)\. Both the synthetic and real\-world experiments demonstrate that*FO\-GS*outperforms the baselines\. We provide additional experimental details in[AppendixA](https://arxiv.org/html/2609.18127#A1)\.

### 6\.1Performance Evaluation on Synthetic Data

In this section, we compare*FO\-GS*with two existing baselines on synthetic data\. The trajectories are generated according to the fractional\-order dynamics in \([2](https://arxiv.org/html/2609.18127#S3.E2)\)\. The ground\-truth parameters𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}andA⋆A\_\{\\star\}are randomly sampled, and the reported mean squared error \(MSE\) is averaged over five randomly generated system instances, with 20 independent rollouts for each instance\.

Varying trajectory horizons\.We evaluate the MSE of the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}and the system matrixA⋆A\_\{\\star\}as functions of the trajectory horizonttby fixing the noise scaleσ\\sigma\. As shown in[Figure2](https://arxiv.org/html/2609.18127#S6.F2),*FO\-GS*outperforms baselines in estimating both𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}andA⋆A\_\{\\star\}across all trajectory lengths\. The improvement is particularly pronounced for shorter horizons, where accurate identification is most challenging\. A plausible reason is that our method directly fits the original fractional\-order model and exploits its structural decomposition, whereas the baselines rely either on a wavelet\-based proxy for estimating the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}or on a truncated lifted\-state approximation for identifying the system matrixA⋆A\_\{\\star\}\. These additional approximation steps can introduce non\-negligible finite\-sample error, especially when the available trajectory is short\. We also observe that the estimation error generally decreases asttincreases, which is consistent with the theoretical predictions in[Theorems1](https://arxiv.org/html/2609.18127#Thmtheorem1)and[2](https://arxiv.org/html/2609.18127#Thmtheorem2)\.

Varying noise scales\.We examines how the MSE of the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}and the system matrixA⋆A\_\{\\star\}varies with the noise scaleσ\\sigmagiven the same horizontt\.[Figure2](https://arxiv.org/html/2609.18127#S6.F2)shows that*FO\-GS*consistently outperforms the two baselines across all noise levels\. As expected, the MSE of all methods increases as the noise level grows, but the*FO\-GS*remains the most robust, likely because it estimates𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}andA⋆A\_\{\\star\}directly from the original fractional\-order model, whereas the baselines incur additional approximation error through wavelet\-based estimation or system truncation\.

Varying grid sizes\.We study how the number of gridsMMaffects the performance of*FO\-GS*\([Figure2](https://arxiv.org/html/2609.18127#S6.F2)\)\. Consistent with[Theorems1](https://arxiv.org/html/2609.18127#Thmtheorem1)and[2](https://arxiv.org/html/2609.18127#Thmtheorem2), the MSE for both the fractional order𝜶\\boldsymbol\{\\alpha\}and the system matrixAAdecreases as the number of grids increases\. Notably,*FO\-GS*outperforms both*FO\-BS*and*FO\-WT*without requiring a large number of grids: it surpasses the baselines in estimating𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}with roughly ten grids and in estimatingA⋆A\_\{\\star\}with roughly five grids\. This demonstrates that*FO\-GS*is not only accurate but also computationally efficient\.

Smallα⋆\\boldsymbol\{\\alpha\}\_\{\\star\}regime\.We further examine the small fractional\-order regime by uniformly sampling𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}from the interval\[0\.01,0\.2\]\[0\.01,0\.2\]for systems withn=10n=10andn=20n=20, and extending the trajectory horizon up tot=25,600t=25\{,\}600\. As shown in[Table1](https://arxiv.org/html/2609.18127#S6.T1), over the shorter horizonst∈\{100,200,300,400\}t\\in\\\{100,200,300,400\\\}, the empirical convergence is slower than the predictedt−1/2t^\{\-1/2\}rate, indicating stronger finite\-sample effects when the fractional orders are small\. As the trajectory length increases, however, the fitted rates become progressively faster\. Specifically,[Table2](https://arxiv.org/html/2609.18127#S6.T2)shows that over the full extended horizon, the rate scales approximately ast−0\.36t^\{\-0\.36\}forn=10n=10andt−0\.38t^\{\-0\.38\}forn=20n=20, while fitting only the larger horizon regimet≥3200t\\geq 3200yields rates of approximatelyt−0\.43t^\{\-0\.43\}andt−0\.42t^\{\-0\.42\}, respectively\. These results show a clear trend toward thet−1/2t^\{\-1/2\}rate predicted by[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)as the trajectory becomes longer\. We emphasize that the sample\-size requirement in[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1), including its explicitαmin−4\\alpha\_\{\\min\}^\{\-4\}dependence, is a sufficient condition for entering the fast rate regime rather than a necessary or optimal threshold\. Thus, the observed finite sample behavior may be better than what is implied by the conservative sufficient condition\.

Figure 2:Comparison of the MSE on synthetic FOLTI system identification\. Shaded regions indicate95%95\\%confidence intervals \(CI\)\.Figure 3:Subject\-level average test one\-step NMSE on the EEG mental\-arithmetic dataset\. Each point corresponds to one subject and is obtained by averaging the window\-level test one\-step NMSE over all non\-overlapping windows from that subject\.Figure 4:Subject\-level average training one\-step NMSE on the EEG mental\-arithmetic dataset\. Each point corresponds to one subject and is obtained by averaging the window\-level training one\-step NMSE over all non\-overlapping windows from that subject\.Table 1:Log–log fitted rates over short horizons\.Table 2:Log–log fitted rates over long horizons in the small𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}regime\.
### 6\.2Performance Evaluation on Real\-World Data

We evaluate*FO\-GS*on an EEG mental\-arithmetic dataset\[data4010014\]comprising artifact\-free recordings from3636subjects, sampled at500500Hz with a Neurocom2323\-channel system and1919electrodes placed obeying the International10/2010/20scheme\. We use the first minute of the serial\-subtraction task, treating each subject’s recording as a1919\-dimensional time series\. The data are segmented into non\-overlapping windows of lengthW=150W=150, with70%70\\%of samples used for training and30%30\\%for testing\. We compare*FO\-GS*against*FO\-BS*and*FO\-WT*using one\-step NMSE\.[Figure3](https://arxiv.org/html/2609.18127#S6.F3)reports subject\-level average test NMSE\. Training errors are similar across methods as shown in[Figure4](https://arxiv.org/html/2609.18127#S6.F4), but test performance differs markedly:*FO\-GS*consistently achieves the lowest NMSE for all subjects, while*FO\-BS*and*FO\-WT*incur higher errors\. This improvement stems from the fractional\-order identification in*FO\-GS*, where accurate estimation of𝜶\\boldsymbol\{\\alpha\}yields better history weighting and prediction\. In contrast,*FO\-BS*uses a fixed finite\-memory approximation and*FO\-WT*estimates the order via a separate wavelet\-based step\. Overall,*FO\-GS*delivers superior predictive performance\.

Learned\-order interpretation\.We also examine the learned fractional orders on the EEG dataset and find clear evidence of non\-integer, channel\-dependent memory\. ForFO\-GS, the median window\-averaged order is 0\.855; 74\.6% of windows have a cross\-channel order range greater than 0\.5, and 72\.8% contain at least one channel withα<0\.1\\alpha<0\.1\. These results suggest substantial heterogeneity in long\-term memory across EEG channels\.

## 7Conclusion

We study the identification of FOLTI systems from a single observed trajectory and propose*FO\-GS*, a simple two\-stage estimator that exploits the diagonal structure of the Grünwald–Letnikov difference operator to decouple the estimation of the fractional order𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}and the system matrixA⋆A\_\{\\star\}row\-wise\. Under the stability assumption, we show that*FO\-GS*admits high\-probability non\-asymptotic error guarantees for recovering both𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}andA⋆A\_\{\\star\}\.*FO\-GS*outperforms existing baselines on both synthetic and EEG data\. These results indicate that direct single\-trajectory identification of FOLTI systems is both statistically analyzable and practically effective despite the non\-Markovian system dynamics\. Future work should focus on reducing grid\-search cost and relaxing the stability assumption\.

## References

## Content

## Appendix AExperiment Details

### A\.1Synthetic Experiments

We generate trajectories from two\-dimensional fractional\-order LTI systems\. For each synthetic system,αi,⋆\\alpha\_\{i,\\star\}are sampled independently and uniformly from\[0\.1,0\.5\]\[0\.1,0\.5\], andA⋆A\_\{\\star\}is generated with eigenvalues sampled uniformly from\[−0\.5,0\.5\]\[\-0\.5,0\.5\]\. We compare*FO\-GS*with two baselines\.*FO\-BS*uses a truncated lifted\-state representation with memory lengthp=40p=40, binary\-search tolerance10−210^\{\-2\}, search interval\[0\.05,0\.55\]\[0\.05,0\.55\], and ridge parameter10−610^\{\-6\}\.*FO\-WT*uses Haar wavelets with minimum level22, maximum level chosen automatically, linear detrending, and ridge parameter10−610^\{\-6\}for the subsequent OLS step\. Unless otherwise specified,*FO\-GS*searches over\[0\.05,0\.55\]\[0\.05,0\.55\]using2020equally spaced grid points per coordinate and ridge parameter10−610^\{\-6\}\. We report the MSE of both𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}andA⋆A\_\{\\star\}, averaged over55matched systems and2020Monte Carlo trials\.

Varying trajectory horizons\.To study the effect of trajectory length, we vary the horizon overt∈\{50,100,150,200,300,400,500\}t\\in\\\{50,100,150,200,300,400,500\\\}while fixing the noise scale toσ=0\.1\\sigma=0\.1\. The initial state is sampled from a zero\-mean Gaussian distribution with standard deviation4\.04\.0\.

Varying noise scales\.To evaluate robustness to process noise, we fix the trajectory horizon att=200t=200and vary the noise scale overσ∈\{0\.02,0\.05,0\.10,0\.15,0\.20,0\.30,0\.40\}\\sigma\\in\\\{0\.02,0\.05,0\.10,0\.15,0\.20,0\.30,0\.40\\\}\. The initial state is sampled from a zero\-mean Gaussian distribution with standard deviation4\.04\.0\.

Varying grid sizes\.To examine the effect of grid resolution in*FO\-GS*, we fixt=100t=100andσ=0\.01\\sigma=0\.01and vary the number of grid points overM∈\{3,5,10,13,16,20,25\}M\\in\\\{3,5,10,13,16,20,25\\\}\. The initial state is sampled from a zero\-mean Gaussian distribution with standard deviation2\.02\.0\.

### A\.2Real\-World Experiments

EEG preprocessing\.We evaluate all methods on a multi\-subject EEG dataset with3636subjects andn=19n=19channels\. Each subject is treated as a multivariate time series\. We split each subject trajectory into non\-overlapping windows of lengthW=150W=150with strideS=150S=150\. For each window, the first70%70\\%of samples are used for training and the remaining30%30\\%for testing, giving105105training samples and4545test samples per window\. The experiments are run on the raw EEG data without additional normalization\.

Hyperparameters\.We fix the search interval\[0\.05,0\.95\]\[0\.05,0\.95\]\.*FO\-GS*uses5050equally spaced grid points for each row\-wise search\.*FO\-BS*uses binary search with tolerance10−210^\{\-2\}and a truncated lifted\-state representation with memory lengthp=40p=40\.*FO\-WT*uses Haar wavelets with linear detrending and estimates the fractional order from a weighted log\-variance regression over wavelet levels2,3,42,3,4\.

Evaluation metric\.We evaluate methods by one\-step prediction\. We report training and test NMSE, whereNMSE=∑t‖x^t−xt‖22∑t‖xt‖22\.\\operatorname\{NMSE\}=\\frac\{\\sum\_\{t\}\\\|\\hat\{x\}\_\{t\}\-x\_\{t\}\\\|\_\{2\}^\{2\}\}\{\\sum\_\{t\}\\\|x\_\{t\}\\\|\_\{2\}^\{2\}\}\.

## Appendix BProof of[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)

### B\.1Controlling In\-Sample Error via Martingale Offset Complexity

We now adapt the offset martingale complexity argument\[pmlr\-v178\-ziemann22a\]to the row\-wise grid\-search estimator used in[Algorithm1](https://arxiv.org/html/2609.18127#alg1)\. The main difference from the idealized continuous empirical risk minimizer \(ERM\) is that the true parameterαi,⋆\\alpha\_\{i,\\star\}need not belong to the finite search grid\. Consequently, an additional discretization term appears in the basic inequality\[pmlr\-v178\-ziemann22a\]\.

For each coordinatei∈\[n\]i\\in\[n\], define the row\-wise empirical loss

L\(i\)​\(αi,ai\):=∑s=0t−1\|Δαi​xs\+1\(i\)−ai​xs\|2,L^\{\(i\)\}\(\\alpha\_\{i\},a\_\{i\}\):=\\sum\_\{s=0\}^\{t\-1\}\\left\|\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}\\right\|^\{2\},whereai∈ℝ1×na\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}denotes theii\-th row ofAA\. For any candidateαi\\alpha\_\{i\}, let

a^i​\(αi\)∈arg⁡minai∈ℝ1×n​L\(i\)​\(αi,ai\)\.\\hat\{a\}\_\{i\}\(\\alpha\_\{i\}\)\\in\\arg\\min\_\{a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}L^\{\(i\)\}\(\\alpha\_\{i\},a\_\{i\}\)\.The row\-search estimator is

α^i∈arg⁡minαi∈𝒜ϵ,i​L\(i\)​\(αi,a^i​\(αi\)\),a^i:=a^i​\(α^i\)\.\\hat\{\\alpha\}\_\{i\}\\in\\arg\\min\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}L^\{\(i\)\}\\bigl\(\\alpha\_\{i\},\\hat\{a\}\_\{i\}\(\\alpha\_\{i\}\)\\bigr\),\\qquad\\hat\{a\}\_\{i\}:=\\hat\{a\}\_\{i\}\(\\hat\{\\alpha\}\_\{i\}\)\.Letαi∘\\alpha\_\{i\}^\{\\circ\}denote the grid point closest to the true parameter:

αi∘∈arg⁡minαi∈𝒜ϵ,i​\|αi−αi,⋆\|\.\\alpha\_\{i\}^\{\\circ\}\\in\\arg\\min\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\.\(13\)In particular, if the grid resolution isϵi\\epsilon\_\{i\}, then

\|αi∘−αi,⋆\|≤ϵi\.\|\\alpha\_\{i\}^\{\\circ\}\-\\alpha\_\{i,\\star\}\|\\leq\\epsilon\_\{i\}\.Define

bs\(i\)​\(αi\):=\(Δαi−Δαi,⋆\)​xs\+1\(i\)\.b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\left\(\\Delta^\{\\alpha\_\{i\}\}\-\\Delta^\{\\alpha\_\{i,\\star\}\}\\right\)x\_\{s\+1\}^\{\(i\)\}\.Since the true dynamics satisfy

Δαi,⋆​xs\+1\(i\)=ai,⋆​xs\+ηs\(i\),\\Delta^\{\\alpha\_\{i,\\star\}\}x\_\{s\+1\}^\{\(i\)\}=a\_\{i,\\star\}x\_\{s\}\+\\eta\_\{s\}^\{\(i\)\},\(14\)we have

Δαi​xs\+1\(i\)−ai​xs=ηs\(i\)\+bs\(i\)​\(αi\)−\(ai−ai,⋆\)​xs\.\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}=\\eta\_\{s\}^\{\(i\)\}\+b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\(a\_\{i\}\-a\_\{i,\\star\}\)x\_\{s\}\.\(15\)
For convenience, define

rs\(i\)​\(αi,Δ​ai\):=bs\(i\)​\(αi\)−Δ​ai​xs,Δ​ai:=ai−ai,⋆,r\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\},\\Delta a\_\{i\}\):=b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\},\\qquad\\Delta a\_\{i\}:=a\_\{i\}\-a\_\{i,\\star\},and

r^s\(i\):=rs\(i\)​\(α^i,a^i−ai,⋆\)\.\\hat\{r\}\_\{s\}^\{\(i\)\}:=r\_\{s\}^\{\(i\)\}\\left\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\right\)\.
###### Lemma 5\(Row\-wise basic inequality\)\.

For everyi∈\[n\]i\\in\[n\],

∑s=0t−1\|r^s\(i\)\|2≤\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\\left\|\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\|^\{2\}\\leq4​∑s=0t−1⟨−ηs\(i\),r^s\(i\)⟩−∑s=0t−1\|r^s\(i\)\|2\\displaystyle 4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\}^\{\(i\)\},\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\|\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\|^\{2\}\+4∑s=0t−1⟨ηs\(i\),bs\(i\)\(αi∘\)⟩\+2∑s=0t−1\|bs\(i\)\(αi∘\)\|2\.\\displaystyle\+4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\\rangle\+2\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\|^\{2\}\.\(16\)

###### Proof\.

By optimality of the row\-wise grid\-search estimator,

L\(i\)​\(α^i,a^i\)\\displaystyle L^\{\(i\)\}\\left\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\\right\)≤L\(i\)​\(αi∘,a^i​\(αi∘\)\)≤L\(i\)​\(αi∘,ai,⋆\)\.\\displaystyle\\leq L^\{\(i\)\}\\left\(\\alpha\_\{i\}^\{\\circ\},\\hat\{a\}\_\{i\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\)\\leq L^\{\(i\)\}\\left\(\\alpha\_\{i\}^\{\\circ\},a\_\{i,\\star\}\\right\)\.Using \([15](https://arxiv.org/html/2609.18127#A2.E15)\), this gives

∑s=0t−1\|ηs\(i\)\+r^s\(i\)\|2≤∑s=0t−1\|ηs\(i\)\+bs\(i\)​\(αi∘\)\|2\.\\sum\_\{s=0\}^\{t\-1\}\\left\|\\eta\_\{s\}^\{\(i\)\}\+\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\|^\{2\}\\leq\\sum\_\{s=0\}^\{t\-1\}\\left\|\\eta\_\{s\}^\{\(i\)\}\+b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\|^\{2\}\.Expanding both sides and cancelling∑s=0t−1\|ηs\(i\)\|2\\sum\_\{s=0\}^\{t\-1\}\|\\eta\_\{s\}^\{\(i\)\}\|^\{2\}yields

∑s=0t−1\|r^s\(i\)\|2≤\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\\left\|\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\|^\{2\}\\leq−2∑s=0t−1⟨ηs\(i\),r^s\(i\)⟩\\displaystyle\-2\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\\eta\_\{s\}^\{\(i\)\},\\hat\{r\}\_\{s\}^\{\(i\)\}\\right\\rangle\+2∑s=0t−1⟨ηs\(i\),bs\(i\)\(αi∘\)⟩\+∑s=0t−1\|bs\(i\)\(αi∘\)\|2\.\\displaystyle\+2\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\\rangle\+\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\|^\{2\}\.\(17\)Multiplying \([17](https://arxiv.org/html/2609.18127#A2.E17)\) by two and subtracting∑s\|r^s\(i\)\|2\\sum\_\{s\}\|\\hat\{r\}\_\{s\}^\{\(i\)\}\|^\{2\}from both sides gives \([16](https://arxiv.org/html/2609.18127#A2.E16)\)\. ∎

The first line of \([16](https://arxiv.org/html/2609.18127#A2.E16)\) can now be controlled by an offset martingale complexity argument\. In particular, since\(α^i,a^i−ai,⋆\)\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\)is an admissible choice,

∑s=0t−1\|r^s\(i\)\|2≤\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\|\\hat\{r\}\_\{s\}^\{\(i\)\}\|^\{2\}\\leqmaxαi∈𝒜ϵ,isupΔ​ai∈ℝ1×n\{4∑s=0t−1⟨−ηs\(i\),bs\(i\)\(αi\)−Δaixs⟩\\displaystyle\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\sup\_\{\\Delta a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}\\Bigg\\\{4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\\rangle−∑s=0t−1\|bs\(i\)\(αi\)−Δaixs\|2\}\+Γt,igrid,\\displaystyle\\hskip 113\.81102pt\-\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\|^\{2\}\\Bigg\\\}\+\\Gamma\_\{t,i\}^\{\\mathrm\{grid\}\},where

Γt,igrid:=4​∑s=0t−1⟨ηs\(i\),bs\(i\)​\(αi∘\)⟩\+2​∑s=0t−1\|bs\(i\)​\(αi∘\)\|2\\Gamma\_\{t,i\}^\{\\mathrm\{grid\}\}:=4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\\rangle\+2\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\|^\{2\}is the additional error induced by the finite grid\.

We next explicitly perform the maximization overΔ​ai\\Delta a\_\{i\}\. Recall from \([9](https://arxiv.org/html/2609.18127#S4.E9)\) that

Xt=\[x0,…,xt−1\],Xt​Xt⊤=∑s=0t−1xs​xs⊤\.X\_\{t\}=\[x\_\{0\},\\ldots,x\_\{t\-1\}\],\\qquad X\_\{t\}X\_\{t\}^\{\\top\}=\\sum\_\{s=0\}^\{t\-1\}x\_\{s\}x\_\{s\}^\{\\top\}\.For a fixedαi\\alpha\_\{i\}, let

ys\(i\)​\(αi\):=bs\(i\)​\(αi\)\+2​ηs\(i\)\.y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\+2\\eta\_\{s\}^\{\(i\)\}\.Then

4​∑s=0t−1⟨−ηs\(i\),bs\(i\)​\(αi\)−Δ​ai​xs⟩−∑s=0t−1\|bs\(i\)​\(αi\)−Δ​ai​xs\|2\\displaystyle 4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\|^\{2\}=−4∑s=0t−1⟨ηs\(i\),bs\(i\)\(αi\)⟩−∑s=0t−1\|bs\(i\)\(αi\)\|2\+2⟨∑s=0t−1ys\(i\)\(αi\)xs⊤,Δai⟩−ΔaiXtXt⊤Δai⊤\.\\displaystyle=\-4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\right\|^\{2\}\+2\\left\\langle\\sum\_\{s=0\}^\{t\-1\}y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)x\_\{s\}^\{\\top\},\\Delta a\_\{i\}\\right\\rangle\-\\Delta a\_\{i\}X\_\{t\}X\_\{t\}^\{\\top\}\\Delta a\_\{i\}^\{\\top\}\.AssumingXt​Xt⊤≻0X\_\{t\}X\_\{t\}^\{\\top\}\\succ 0, the maximizer is

Δ​ai∗​\(αi\)=\(∑s=0t−1ys\(i\)​\(αi\)​xs⊤\)​\(Xt​Xt⊤\)−1\.\\Delta a\_\{i\}^\{\*\}\(\\alpha\_\{i\}\)=\\left\(\\sum\_\{s=0\}^\{t\-1\}y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)x\_\{s\}^\{\\top\}\\right\)\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\.Therefore,

supΔ​ai\{4​∑s=0t−1⟨−ηs\(i\),bs\(i\)​\(αi\)−Δ​ai​xs⟩−∑s=0t−1\|bs\(i\)​\(αi\)−Δ​ai​xs\|2\}\\displaystyle\\sup\_\{\\Delta a\_\{i\}\}\\Bigg\\\{4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\|^\{2\}\\Bigg\\\}=Ut,i​\(αi\)\+Vt,i​\(αi\),\\displaystyle=U\_\{t,i\}\(\\alpha\_\{i\}\)\+V\_\{t,i\}\(\\alpha\_\{i\}\),where

Ut,i​\(αi\):=4​∑s=0t−1⟨−ηs\(i\),bs\(i\)​\(αi\)⟩−∑s=0t−1\|bs\(i\)​\(αi\)\|2,U\_\{t,i\}\(\\alpha\_\{i\}\):=4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\}^\{\(i\)\},b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\right\|^\{2\},and

Vt,i\(αi\):=‖\(∑s=0t−1ys\(i\)\(αi\)xs⊤\)\(XtXt⊤\)−1/2‖22\.V\_\{t,i\}\(\\alpha\_\{i\}\):=\\left\\\|\\left\(\\sum\_\{s=0\}^\{t\-1\}y\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)x\_\{s\}^\{\\top\}\\right\)\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{2\}^\{2\}\.Combining everything together gives the following row\-wise in\-sample error bound:

∑s=0t−1\|bs\(i\)​\(α^i\)−\(a^i−ai,⋆\)​xs\|2≤maxαi∈𝒜ϵ,i⁡\{Ut,i​\(αi\)\+Vt,i​\(αi\)\}\+Γt,igrid\.\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\hat\{\\alpha\}\_\{i\}\)\-\(\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\)x\_\{s\}\\right\|^\{2\}\\leq\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\left\\\{U\_\{t,i\}\(\\alpha\_\{i\}\)\+V\_\{t,i\}\(\\alpha\_\{i\}\)\\right\\\}\+\\Gamma\_\{t,i\}^\{\\mathrm\{grid\}\}\.
Finally, summing over all coordinatesi∈\[n\]i\\in\[n\]gives

∑s=0t−1‖\(Δ𝜶^−Δ𝜶⋆\)​xs\+1−\(A^−A⋆\)​xs‖22≤∑i=1nmaxαi∈𝒜ϵ,i⁡\{Ut,i​\(αi\)\+Vt,i​\(αi\)\}\+Γtgrid,\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\\left\\\|\\left\(\\Delta^\{\\hat\{\\boldsymbol\{\\alpha\}\}\}\-\\Delta^\{\\boldsymbol\{\\alpha\}^\{\\star\}\}\\right\)x\_\{s\+1\}\-\(\\hat\{A\}\-A\_\{\\star\}\)x\_\{s\}\\right\\\|\_\{2\}^\{2\}\\leq\\sum\_\{i=1\}^\{n\}\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}\\left\\\{U\_\{t,i\}\(\\alpha\_\{i\}\)\+V\_\{t,i\}\(\\alpha\_\{i\}\)\\right\\\}\+\\Gamma\_\{t\}^\{\\mathrm\{grid\}\},where

Γtgrid:=∑i=1nΓt,igrid\.\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}:=\\sum\_\{i=1\}^\{n\}\\Gamma\_\{t,i\}^\{\\mathrm\{grid\}\}\.Equivalently, letting

𝜶∘:=\(α1∘,…,αn∘\),\\boldsymbol\{\\alpha\}^\{\\circ\}:=\(\\alpha\_\{1\}^\{\\circ\},\\ldots,\\alpha\_\{n\}^\{\\circ\}\),and defining the matrices

Bt​\(𝜶∘\):=\[b0​\(𝜶∘\),⋯,bt−1​\(𝜶∘\)\],Wt:=\[η0,⋯,ηt−1\],B\_\{t\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\):=\\begin\{bmatrix\}b\_\{0\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\),&\\cdots,&b\_\{t\-1\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\)\\end\{bmatrix\},\\qquad W\_\{t\}:=\\begin\{bmatrix\}\\eta\_\{0\},&\\cdots,&\\eta\_\{t\-1\}\\end\{bmatrix\},the grid\-discretization contribution can be written compactly as

Γtgrid=4​⟨Wt,Bt​\(𝜶∘\)⟩F\+2​‖Bt​\(𝜶∘\)‖F2\.\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}=4\\left\\langle W\_\{t\},B\_\{t\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\)\\right\\rangle\_\{F\}\+2\\left\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\)\\right\\\|\_\{F\}^\{2\}\.
###### Lemma 6\(Parameterized bound for the offset term\)\.

Fix𝛂\\boldsymbol\{\\alpha\}and assumeXt​Xt⊤≻0X\_\{t\}X\_\{t\}^\{\\top\}\\succ 0\. For an arbitrary auxiliary matrix perturbationΔ​A∈ℝn×n\\Delta A\\in\\mathbb\{R\}^\{n\\times n\}, whereΔ​A:=A−A⋆\\Delta A:=A\-A\_\{\\star\}, consider

Φ𝜶​\(Δ​A\)=4​∑s=0t−1⟨−ηs,bs​\(𝜶\)−Δ​A​xs⟩−∑s=0t−1‖bs​\(𝜶\)−Δ​A​xs‖22\.\\Phi\_\{\\boldsymbol\{\\alpha\}\}\(\\Delta A\)=4\\sum\_\{s=0\}^\{t\-1\}\\left\\langle\-\\eta\_\{s\},b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\-\\Delta Ax\_\{s\}\\right\\rangle\-\\sum\_\{s=0\}^\{t\-1\}\\left\\\|b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\-\\Delta Ax\_\{s\}\\right\\\|\_\{2\}^\{2\}\.\(18\)
For anyτ∈\(0,1\]\\tau\\in\(0,1\], define

Ut,τ​\(𝜶\):=−4​⟨Wt,Bt​\(𝜶\)⟩F−τ​‖Bt​\(𝜶\)‖F2\.U\_\{t,\\tau\}\(\\boldsymbol\{\\alpha\}\):=\-4\\langle W\_\{t\},B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\rangle\_\{F\}\-\\tau\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}\.\(19\)Then

Ut​\(𝜶\)=Ut,τ​\(𝜶\)−\(1−τ\)​‖Bt​\(𝜶\)‖F2\.U\_\{t\}\(\\boldsymbol\{\\alpha\}\)=U\_\{t,\\tau\}\(\\boldsymbol\{\\alpha\}\)\-\(1\-\\tau\)\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}\.\(20\)Moreover, for every fixed deterministic𝛂\\boldsymbol\{\\alpha\}andδ∈\(0,1\)\\delta\\in\(0,1\),

Pr⁡\(Ut,τ​\(𝜶\)≤8​σ2τ​log⁡1δ\)≥1−δ\.\\Pr\\left\(U\_\{t,\\tau\}\(\\boldsymbol\{\\alpha\}\)\\leq\\frac\{8\\sigma^\{2\}\}\{\\tau\}\\log\\frac\{1\}\{\\delta\}\\right\)\\geq 1\-\\delta\.\(21\)For the row\-wise grid search, define

Ut,τ,i\(αi\):=−4∑s=0t−1ηs\(i\)bs\(i\)\(αi\)−τ∑s=0t−1\|bs\(i\)\(αi\)\|2\.U\_\{t,\\tau,i\}\(\\alpha\_\{i\}\):=\-4\\sum\_\{s=0\}^\{t\-1\}\\eta\_\{s\}^\{\(i\)\}b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\tau\\sum\_\{s=0\}^\{t\-1\}\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\.\(22\)Then, with probability at least1−δ1\-\\delta, simultaneously for alli∈\[n\]i\\in\[n\],

maxαi∈𝒜ϵ,i⁡Ut,τ,i​\(αi\)≤8​σ2τ​log⁡\(Mi​nδ\)\.\\max\_\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\}U\_\{t,\\tau,i\}\(\\alpha\_\{i\}\)\\leq\\frac\{8\\sigma^\{2\}\}\{\\tau\}\\log\\left\(\\frac\{M\_\{i\}n\}\{\\delta\}\\right\)\.\(23\)

###### Proof\.

Expanding \([18](https://arxiv.org/html/2609.18127#A2.E18)\) gives

Φ𝜶​\(Δ​A\)=\\displaystyle\\Phi\_\{\\boldsymbol\{\\alpha\}\}\(\\Delta A\)=\{\}−4​⟨Wt,Bt​\(𝜶\)⟩F−‖Bt​\(𝜶\)‖F2\\displaystyle\-4\\langle W\_\{t\},B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\rangle\_\{F\}\-\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}\+2​⟨\(Bt​\(𝜶\)\+2​Wt\)​Xt⊤,Δ​A⟩F−tr⁡\(Δ​A​Xt​Xt⊤​Δ​A⊤\)\.\\displaystyle\+2\\left\\langle\\bigl\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+2W\_\{t\}\\bigr\)X\_\{t\}^\{\\top\},\\Delta A\\right\\rangle\_\{F\}\-\\operatorname\{tr\}\\\!\\left\(\\Delta AX\_\{t\}X\_\{t\}^\{\\top\}\\Delta A^\{\\top\}\\right\)\.\(24\)The maximizer of this auxiliary offset objective is

Δ​Aopt=\(Bt​\(𝜶\)\+2​Wt\)​Xt⊤​\(Xt​Xt⊤\)−1\.\\Delta A\_\{\\mathrm\{opt\}\}=\\bigl\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+2W\_\{t\}\\bigr\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\.Substitution gives

supΔ​AΦ𝜶​\(Δ​A\)=Ut​\(𝜶\)\+Vt​\(𝜶\),\\sup\_\{\\Delta A\}\\Phi\_\{\\boldsymbol\{\\alpha\}\}\(\\Delta A\)=U\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+V\_\{t\}\(\\boldsymbol\{\\alpha\}\),where

Ut​\(𝜶\)=−4​⟨Wt,Bt​\(𝜶\)⟩F−‖Bt​\(𝜶\)‖F2,U\_\{t\}\(\\boldsymbol\{\\alpha\}\)=\-4\\langle W\_\{t\},B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\rangle\_\{F\}\-\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\},and

Vt\(𝜶\)=‖\(Bt\(𝜶\)\+2Wt\)Xt⊤\(XtXt⊤\)−1/2‖F2\.V\_\{t\}\(\\boldsymbol\{\\alpha\}\)=\\left\\\|\\bigl\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+2W\_\{t\}\\bigr\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}^\{2\}\.Then \([20](https://arxiv.org/html/2609.18127#A2.E20)\) follows directly from the definitions\.

For fixed deterministic𝜶\\boldsymbol\{\\alpha\},bs​\(𝜶\)b\_\{s\}\(\\boldsymbol\{\\alpha\}\)isℱs\\mathcal\{F\}\_\{s\}\-measurable\. Hence, forλ\>0\\lambda\>0,

𝔼⁡\[exp⁡\(λ⁡\[−4​⟨ηs,bs​\(𝜶\)⟩−τ​‖bs​\(𝜶\)‖22\]\)\|ℱs\]=exp⁡\(\(8​σ2​λ2−τ​λ\)​‖bs​\(𝜶\)‖22\)\.\\displaystyle\\mathbb\{E\}\\left\[\\exp\\left\(\\lambda\\left\[\-4\\langle\\eta\_\{s\},b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\\rangle\-\\tau\\\|b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{2\}^\{2\}\\right\]\\right\)\\middle\|\\mathcal\{F\}\_\{s\}\\right\]=\\exp\\left\(\(8\\sigma^\{2\}\\lambda^\{2\}\-\\tau\\lambda\)\\\|b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{2\}^\{2\}\\right\)\.The right\-hand side is at most one whenever0<λ≤τ/\(8​σ2\)0<\\lambda\\leq\\tau/\(8\\sigma^\{2\}\)\. Iterating conditional expectations, takingλ=τ/\(8​σ2\)\\lambda=\\tau/\(8\\sigma^\{2\}\), and applying Markov’s inequality proves \([21](https://arxiv.org/html/2609.18127#A2.E21)\)\. The row\-wise statement follows by taking failure probabilityδ/\(n​Mi\)\\delta/\(nM\_\{i\}\)for each fixed grid point and applying a union bound over all rows and grid points\. ∎

###### Lemma 7\(High\-probability bound forVt​\(𝜶\)V\_\{t\}\(\\boldsymbol\{\\alpha\}\)\)\.

Define

𝔯t\(δ,k\):=Ct​λmin​\(Γk\)\(nlognδ\+logdet\(ΓtΓk−1\)\)1/2,\\mathfrak\{r\}\_\{t\}\(\\delta,k\):=\\frac\{C\}\{\\sqrt\{t\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\}\\left\(n\\log\\frac\{n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)^\{1/2\},whereC\>0C\>0is the universal constant in[Lemma3](https://arxiv.org/html/2609.18127#Thmlemma3)\. Suppose that

tk≥c\(nlognδ\+logdet\(ΓtΓk−1\)\)\.\\frac\{t\}\{k\}\\geq c\\left\(n\\log\\frac\{n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)\.For the row\-wise search intervals𝒜i=\[α¯i,α¯i\]\\mathcal\{A\}\_\{i\}=\[\\underline\{\\alpha\}\_\{i\},\\overline\{\\alpha\}\_\{i\}\], define

S1,i:=supu∈𝒜i∑j≥1\|∂uψ⁡\(u,j\)\|,S1:=max1≤i≤n⁡S1,i\.S\_\{1,i\}:=\\sup\_\{u\\in\\mathcal\{A\}\_\{i\}\}\\sum\_\{j\\geq 1\}\|\\partial\_\{u\}\\psi\(u,j\)\|,\\qquad S\_\{1\}:=\\max\_\{1\\leq i\\leq n\}S\_\{1,i\}\.\(25\)Then, with probability at least1−δ1\-\\delta, simultaneously for all𝛂\\boldsymbol\{\\alpha\}in the search set,

Vt​\(𝜶\)≤tr⁡\(Xt​Xt⊤\)​\(S1​‖𝜶−𝜶⋆‖∞\+2​𝔯t​\(δ,k\)\)2\.V\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\leq\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\\left\(S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\+2\\mathfrak\{r\}\_\{t\}\(\\delta,k\)\\right\)^\{2\}\.\(26\)In particular,

Vt​\(𝜶\)≤\\displaystyle V\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\leq\{\}2​S12​‖𝜶−𝜶⋆‖∞2​tr⁡\(Xt​Xt⊤\)\\displaystyle 2S\_\{1\}^\{2\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\+8​C2​tr⁡\(Xt​Xt⊤\)t​λmin​\(Γk\)\(nlognδ\+logdet\(ΓtΓk−1\)\)\.\\displaystyle\+\\frac\{8C^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\{t\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\left\(n\\log\\frac\{n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)\.

###### Proof\.

Since

∑s=0t−1\(bs​\(𝜶\)\+2​ηs\)​xs⊤=\(Bt​\(𝜶\)\+2​Wt\)​Xt⊤,\\sum\_\{s=0\}^\{t\-1\}\\bigl\(b\_\{s\}\(\\boldsymbol\{\\alpha\}\)\+2\\eta\_\{s\}\\bigr\)x\_\{s\}^\{\\top\}=\\bigl\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\+2W\_\{t\}\\bigr\)X\_\{t\}^\{\\top\},we have

Vt\(𝜶\)1/2≤‖Bt\(𝜶\)Xt⊤\(XtXt⊤\)−1/2‖F\+2‖WtXt⊤\(XtXt⊤\)−1/2‖F\.\\displaystyle V\_\{t\}\(\\boldsymbol\{\\alpha\}\)^\{1/2\}\\leq\\left\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}\+2\\left\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}\.\(27\)Let

PX:=Xt⊤​\(Xt​Xt⊤\)−1​Xt\.P\_\{X\}:=X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}X\_\{t\}\.ThenPXP\_\{X\}is an orthogonal projection, so

‖Bt\(𝜶\)Xt⊤\(XtXt⊤\)−1/2‖F2=tr\(Bt\(𝜶\)PXBt\(𝜶\)⊤\)≤∥Bt\(𝜶\)∥F2\.\\displaystyle\\left\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}^\{2\}=\\operatorname\{tr\}\\\!\\left\(B\_\{t\}\(\\boldsymbol\{\\alpha\}\)P\_\{X\}B\_\{t\}\(\\boldsymbol\{\\alpha\}\)^\{\\top\}\\right\)\\leq\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}\.The convolution bound established in the proof of[Lemma4](https://arxiv.org/html/2609.18127#Thmlemma4)gives

‖Bt​\(𝜶\)‖F2≤S12​‖𝜶−𝜶⋆‖∞2​tr⁡\(Xt​Xt⊤\)\.\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}\\leq S\_\{1\}^\{2\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\.Hence

‖Bt\(𝜶\)Xt⊤\(XtXt⊤\)−1/2‖F≤S1∥𝜶−𝜶⋆∥∞tr⁡\(Xt​Xt⊤\)\.\\left\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}\\leq S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\sqrt\{\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\.\(28\)Moreover,

WtXt⊤\(XtXt⊤\)−1/2=\(WtXt⊤\(XtXt⊤\)−1\)\(XtXt⊤\)1/2,W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}=\\left\(W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\right\)\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{1/2\},and therefore[Lemma3](https://arxiv.org/html/2609.18127#Thmlemma3)implies

‖WtXt⊤\(XtXt⊤\)−1/2‖F≤𝔯t\(δ,k\)tr⁡\(Xt​Xt⊤\)\\left\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\right\\\|\_\{F\}\\leq\\mathfrak\{r\}\_\{t\}\(\\delta,k\)\\sqrt\{\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\(29\)with probability at least1−δ1\-\\delta\. Substituting \([28](https://arxiv.org/html/2609.18127#A2.E28)\) and \([29](https://arxiv.org/html/2609.18127#A2.E29)\) into \([27](https://arxiv.org/html/2609.18127#A2.E27)\) proves \([26](https://arxiv.org/html/2609.18127#A2.E26)\); the expanded form follows from\(a\+b\)2≤2​a2\+2​b2\(a\+b\)^\{2\}\\leq 2a^\{2\}\+2b^\{2\}\. ∎

###### Lemma 8\(Absorbable high\-probability in\-sample error bound\)\.

Forτ∈\(0,1\]\\tau\\in\(0,1\]andρ\>0\\rho\>0, suppose that

tk≥c\(nlog3​nδ\+logdet\(ΓtΓk−1\)\)\.\\frac\{t\}\{k\}\\geq c\\left\(n\\log\\frac\{3n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)\.Then, with probability at least1−δ1\-\\delta,

∑s=0t−1‖\(Δ𝜶^−Δ𝜶⋆\)​xs\+1−\(A^−A⋆\)​xs‖22\\displaystyle\\sum\_\{s=0\}^\{t\-1\}\\left\\\|\\left\(\\Delta^\{\\hat\{\\boldsymbol\{\\alpha\}\}\}\-\\Delta^\{\\boldsymbol\{\\alpha\}\_\{\\star\}\}\\right\)x\_\{s\+1\}\-\(\\hat\{A\}\-A\_\{\\star\}\)x\_\{s\}\\right\\\|\_\{2\}^\{2\}≤\(τ\+ρ\)​S12​‖𝜶^−𝜶⋆‖∞2​tr⁡\(Xt​Xt⊤\)\+3​S12​ϵmax2​tr⁡\(Xt​Xt⊤\)\\displaystyle\\quad\\leq\(\\tau\+\\rho\)S\_\{1\}^\{2\}\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\+3S\_\{1\}^\{2\}\\epsilon\_\{\\max\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\+4​\(1\+ρ−1\)​C2​tr⁡\(Xt​Xt⊤\)t​λmin​\(Γk\)\(nlog3​nδ\+logdet\(ΓtΓk−1\)\)\\displaystyle\\qquad\+\\frac\{4\(1\+\\rho^\{\-1\}\)C^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\{t\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\left\(n\\log\\frac\{3n\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)\+8​σ2τ∑i=1nlog\(3​Mi​nδ\)\+8σ2log\(3δ\)\.\\displaystyle\\qquad\+\\frac\{8\\sigma^\{2\}\}\{\\tau\}\\sum\_\{i=1\}^\{n\}\\log\\left\(\\frac\{3M\_\{i\}n\}\{\\delta\}\\right\)\+8\\sigma^\{2\}\\log\\left\(\\frac\{3\}\{\\delta\}\\right\)\.\(30\)

###### Proof\.

For each row, recall

Δ​ai:=a^i−ai,⋆,r^s\(i\):=bs\(i\)​\(α^i\)−Δ​ai​xs\.\\Delta a\_\{i\}:=\\hat\{a\}\_\{i\}\-a\_\{i,\\star\},\\qquad\\hat\{r\}\_\{s\}^\{\(i\)\}:=b\_\{s\}^\{\(i\)\}\(\\hat\{\\alpha\}\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\.The row\-wise basic inequality in[Lemma5](https://arxiv.org/html/2609.18127#Thmlemma5)and the quadratic maximization in Lemma[6](https://arxiv.org/html/2609.18127#Thmlemma6)give

∑s=0t−1\|r^s\(i\)\|2≤Ut,i​\(α^i\)\+Vt,i​\(α^i\)\+Γt,igrid\.\\sum\_\{s=0\}^\{t\-1\}\|\\hat\{r\}\_\{s\}^\{\(i\)\}\|^\{2\}\\leq U\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\+V\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\+\\Gamma\_\{t,i\}^\{\\mathrm\{grid\}\}\.Summing over rows,

ℰt:=∑s=0t−1‖\(Δ𝜶^−Δ𝜶⋆\)​xs\+1−\(A^−A⋆\)​xs‖22≤Ut​\(𝜶^\)\+Vt​\(𝜶^\)\+Γtgrid\.\\displaystyle\\mathcal\{E\}\_\{t\}:=\\sum\_\{s=0\}^\{t\-1\}\\left\\\|\\left\(\\Delta^\{\\hat\{\\boldsymbol\{\\alpha\}\}\}\-\\Delta^\{\\boldsymbol\{\\alpha\}\_\{\\star\}\}\\right\)x\_\{s\+1\}\-\(\\hat\{A\}\-A\_\{\\star\}\)x\_\{s\}\\right\\\|\_\{2\}^\{2\}\\leq U\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\+V\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\+\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}\.\(31\)By \([20](https://arxiv.org/html/2609.18127#A2.E20)\),

Ut​\(𝜶^\)=Ut,τ​\(𝜶^\)−\(1−τ\)​‖Bt​\(𝜶^\)‖F2\.U\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)=U\_\{t,\\tau\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\-\(1\-\\tau\)\\\|B\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\\\|\_\{F\}^\{2\}\.Young’s inequality and the projection inequality give

Ut\(𝜶^\)\+Vt\(𝜶^\)≤Ut,τ\(𝜶^\)\+\(τ\+ρ\)∥Bt\(𝜶^\)∥F2\+4\(1\+ρ−1\)∥WtXt⊤\(XtXt⊤\)−1/2∥F2\.\\displaystyle U\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\+V\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\\leq U\_\{t,\\tau\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\+\(\\tau\+\\rho\)\\\|B\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\\\|\_\{F\}^\{2\}\+4\(1\+\\rho^\{\-1\}\)\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\\|\_\{F\}^\{2\}\.Lemma[6](https://arxiv.org/html/2609.18127#Thmlemma6), with failure probabilityδ/3\\delta/3, gives

Ut,τ​\(𝜶^\)≤8​σ2τ​∑i=1nlog⁡\(3​Mi​nδ\)\.U\_\{t,\\tau\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\\leq\\frac\{8\\sigma^\{2\}\}\{\\tau\}\\sum\_\{i=1\}^\{n\}\\log\\left\(\\frac\{3M\_\{i\}n\}\{\\delta\}\\right\)\.Also, the convolution bound established in the proof of[Lemma4](https://arxiv.org/html/2609.18127#Thmlemma4)gives

‖Bt​\(𝜶^\)‖F2≤S12​‖𝜶^−𝜶⋆‖∞2​tr⁡\(Xt​Xt⊤\)\.\\\|B\_\{t\}\(\\hat\{\\boldsymbol\{\\alpha\}\}\)\\\|\_\{F\}^\{2\}\\leq S\_\{1\}^\{2\}\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\.
Next, retain the full good event used in the proof of[Lemma3](https://arxiv.org/html/2609.18127#Thmlemma3)\. With failure probabilityδ/3\\delta/3, this event simultaneously gives

‖Wt​Xt⊤​\(Xt​Xt⊤\)−1‖op≤𝔯t​\(δ3,k\)\\left\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq\\mathfrak\{r\}\_\{t\}\\\!\\left\(\\frac\{\\delta\}\{3\},k\\right\)and the complement of the eventℰ3\\mathcal\{E\}\_\{3\}from that proof, namely

Xt​Xt⊤⪯t​Γ¯t​\(δ3\),Γ¯t​\(δ0\):=σ2​nδ0​Γt\.X\_\{t\}X\_\{t\}^\{\\top\}\\preceq t\\,\\overline\{\\Gamma\}\_\{t\}\\\!\\left\(\\frac\{\\delta\}\{3\}\\right\),\\qquad\\overline\{\\Gamma\}\_\{t\}\(\\delta\_\{0\}\):=\\frac\{\\sigma^\{2\}n\}\{\\delta\_\{0\}\}\\Gamma\_\{t\}\.\(32\)Consequently,

∥WtXt⊤\(XtXt⊤\)−1/2∥F2\\displaystyle\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1/2\}\\\|\_\{F\}^\{2\}≤𝔯t2​\(δ3,k\)​tr⁡\(Xt​Xt⊤\)\.\\displaystyle\\leq\\mathfrak\{r\}\_\{t\}^\{2\}\\\!\\left\(\\frac\{\\delta\}\{3\},k\\right\)\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\.Thus the upper empirical state energy control in \([32](https://arxiv.org/html/2609.18127#A2.E32)\) does not require a fourth event or an additional allocation of the failure probability\.

Finally, letBt∘:=Bt​\(𝜶∘\)B\_\{t\}^\{\\circ\}:=B\_\{t\}\(\\boldsymbol\{\\alpha\}^\{\\circ\}\)\. Then

Γtgrid=\(4​⟨Wt,Bt∘⟩F−‖Bt∘‖F2\)\+3​‖Bt∘‖F2\.\\Gamma\_\{t\}^\{\\mathrm\{grid\}\}=\\left\(4\\langle W\_\{t\},B\_\{t\}^\{\\circ\}\\rangle\_\{F\}\-\\\|B\_\{t\}^\{\\circ\}\\\|\_\{F\}^\{2\}\\right\)\+3\\\|B\_\{t\}^\{\\circ\}\\\|\_\{F\}^\{2\}\.The same conditional moment generating function argument in[Lemma6](https://arxiv.org/html/2609.18127#Thmlemma6), with failure probabilityδ/3\\delta/3, yields

4​⟨Wt,Bt∘⟩F−‖Bt∘‖F2≤8​σ2​log⁡\(3δ\),4\\langle W\_\{t\},B\_\{t\}^\{\\circ\}\\rangle\_\{F\}\-\\\|B\_\{t\}^\{\\circ\}\\\|\_\{F\}^\{2\}\\leq 8\\sigma^\{2\}\\log\\left\(\\frac\{3\}\{\\delta\}\\right\),while the convolution bound gives

‖Bt∘‖F2≤S12​ϵmax2​tr⁡\(Xt​Xt⊤\)\.\\\|B\_\{t\}^\{\\circ\}\\\|\_\{F\}^\{2\}\\leq S\_\{1\}^\{2\}\\epsilon\_\{\\max\}^\{2\}\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\.Substituting these three bounds into \([31](https://arxiv.org/html/2609.18127#A2.E31)\) and applying a union bound proves \([30](https://arxiv.org/html/2609.18127#A2.E30)\) and \([32](https://arxiv.org/html/2609.18127#A2.E32)\)\. ∎

### B\.2Lower Isometry for the Row\-Wise Grid\-Search Estimator

The row\-wise structure ofFO\-GSallows the lower isometry analysis to be carried out coordinate\-wise\. For each rowii, we profile out the corresponding row ofAAand study the resulting noiseless prediction error as a function ofαi\\alpha\_\{i\}\. This quantity captures the curvature in the fractional\-order parameter and will be used to relate the in\-sample prediction error to the estimation error inαi\\alpha\_\{i\}\.

For eachi∈\[n\]i\\in\[n\], recall

bs\(i\)​\(αi\):=\(Δαi−Δαi,⋆\)​xs\+1\(i\),Δ​ai:=ai−ai,⋆,b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\left\(\\Delta^\{\\alpha\_\{i\}\}\-\\Delta^\{\\alpha\_\{i,\\star\}\}\\right\)x\_\{s\+1\}^\{\(i\)\},\\qquad\\Delta a\_\{i\}:=a\_\{i\}\-a\_\{i,\\star\},and define

Qt,i​\(αi,Δ​ai\):=∑s=0t−1\|bs\(i\)​\(αi\)−Δ​ai​xs\|2\.Q\_\{t,i\}\(\\alpha\_\{i\},\\Delta a\_\{i\}\):=\\sum\_\{s=0\}^\{t\-1\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\|^\{2\}\.SinceFO\-GSprofiles outaia\_\{i\}for every candidateαi\\alpha\_\{i\}, it is natural to introduce the profiled noiseless error

Q¯t,i​\(αi\):=infΔ​ai∈ℝ1×nQt,i​\(αi,Δ​ai\)\.\\underline\{Q\}\_\{t,i\}\(\\alpha\_\{i\}\):=\\inf\_\{\\Delta a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}Q\_\{t,i\}\(\\alpha\_\{i\},\\Delta a\_\{i\}\)\.In particular,

Qt,i​\(α^i,a^i−ai,⋆\)≥Q¯t,i​\(α^i\)\.Q\_\{t,i\}\\left\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\right\)\\geq\\underline\{Q\}\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\.
###### Lemma 9\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds\. For eachi∈\[n\]i\\in\[n\], define

α¯i,loc\\displaystyle\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\}:=αi,⋆2,\\displaystyle:=\\frac\{\\alpha\_\{i,\\star\}\}\{2\},ℨi\\displaystyle\\mathfrak\{Z\}\_\{i\}:=ζ⁡\(1\+αi,⋆2\)−2​ζ′​\(1\+αi,⋆2\)\+ζ′′​\(1\+αi,⋆2\),\\displaystyle:=\\zeta\\\!\\left\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\\right\)\-2\\zeta^\{\\prime\}\\\!\\left\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\\right\)\+\\zeta^\{\\prime\\prime\}\\\!\\left\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\\right\),Ki\\displaystyle K\_\{i\}:=9​e4​σ​C~G​ζ⁡\(2\)​ℨi\.\\displaystyle:=\\frac\{9e\}\{4\}\\sigma\\widetilde\{C\}\_\{G\}\\sqrt\{\\zeta\(2\)\}\\,\\mathfrak\{Z\}\_\{i\}\.Then, for everyαi∈\[α¯i,loc,1\]\\alpha\_\{i\}\\in\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\],

1t​∑s=0t−1𝔼​\|bs\(i\)​\(αi\)−\(αi−αi,⋆\)​∑j≥1∂αψ⁡\(αi,⋆,j\)​xs\+1−j\(i\)\|2≤Ki24​\|αi−αi,⋆\|4\.\\displaystyle\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\(\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\)\\sum\_\{j\\geq 1\}\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\)x\_\{s\+1\-j\}^\{\(i\)\}\\right\|^\{2\}\\leq\\frac\{K\_\{i\}^\{2\}\}\{4\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{4\}\.

###### Proof\.

For rowii, let

S2:=supu∈\[a,1\]∑j=1∞\|∂u2ψ⁡\(u,j\)\|,a\>0S\_\{2\}:=\\sup\_\{u\\in\[a,1\]\}\\sum\_\{j=1\}^\{\\infty\}\|\\partial\_\{u\}^\{2\}\\psi\(u,j\)\|,\\qquad a\>0and

ρs\(i\)​\(α\):=bs\(i\)​\(α\)−\(α−αi,⋆\)​∑j=1s\+1∂αψ⁡\(αi,⋆,j\)​xs\+1−j\(i\)\.\\rho\_\{s\}^\{\(i\)\}\(\\alpha\):=b\_\{s\}^\{\(i\)\}\(\\alpha\)\-\(\\alpha\-\\alpha\_\{i,\\star\}\)\\sum\_\{j=1\}^\{s\+1\}\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\)x\_\{s\+1\-j\}^\{\(i\)\}\.Taylor’s theorem in integral form gives

ψ⁡\(α,j\)−ψ⁡\(αi,⋆,j\)−\(α−αi,⋆\)​∂αψ⁡\(αi,⋆,j\)=\(α−αi,⋆\)2​∫01\(1−τ\)​∂α2ψ⁡\(αi,⋆\+τ⁡\(α−αi,⋆\),j\)​𝑑τ\.\\psi\(\\alpha,j\)\-\\psi\(\\alpha\_\{i,\\star\},j\)\-\(\\alpha\-\\alpha\_\{i,\\star\}\)\\,\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\)=\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{2\}\\int\_\{0\}^\{1\}\(1\-\\tau\)\\partial\_\{\\alpha\}^\{2\}\\psi\(\\alpha\_\{i,\\star\}\+\\tau\(\\alpha\-\\alpha\_\{i,\\star\}\),j\)\\,d\\tau\.Therefore,

ρs\(i\)​\(α\)=\(α−αi,⋆\)2​∫01\(1−τ\)​∑j=1s\+1∂α2ψ⁡\(αi,⋆\+τ⁡\(α−αi,⋆\),j\)​xs\+1−j\(i\)​𝑑τ\.\\rho\_\{s\}^\{\(i\)\}\(\\alpha\)=\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{2\}\\int\_\{0\}^\{1\}\(1\-\\tau\)\\sum\_\{j=1\}^\{s\+1\}\\partial\_\{\\alpha\}^\{2\}\\psi\(\\alpha\_\{i,\\star\}\+\\tau\(\\alpha\-\\alpha\_\{i,\\star\}\),j\)x\_\{s\+1\-j\}^\{\(i\)\}\\,d\\tau\.Forj≥2j\\geq 2, letψ⁡\(u,j\)=−u⁡\(1−u\)j​Qj​\(u\),\\psi\(u,j\)=\-\\frac\{u\(1\-u\)\}\{j\}Q\_\{j\}\(u\),andQj​\(u\):=∏m=2j−1\(1−um\)\.Q\_\{j\}\(u\):=\\prod\_\{m=2\}^\{j\-1\}\\left\(1\-\\frac\{u\}\{m\}\\right\)\.

Define

H1,j​\(u\)=∑m=2j−11m−u,H2,j​\(u\)=∑m=2j−11\(m−u\)2\.H\_\{1,j\}\(u\)=\\sum\_\{m=2\}^\{j\-1\}\\frac\{1\}\{m\-u\},\\qquad H\_\{2,j\}\(u\)=\\sum\_\{m=2\}^\{j\-1\}\\frac\{1\}\{\(m\-u\)^\{2\}\}\.
Then

Qj′​\(u\)=−Qj​\(u\)​H1,j​\(u\),Q\_\{j\}^\{\\prime\}\(u\)=\-Q\_\{j\}\(u\)H\_\{1,j\}\(u\),and

\|Qj′′​\(u\)\|=Qj​\(u\)​\(H1,j​\(u\)2−H2,j​\(u\)\)≤Qj​\(u\)​H1,j​\(u\)2\.\\left\|Q\_\{j\}^\{\\prime\\prime\}\(u\)\\right\|=Q\_\{j\}\(u\)\\left\(H\_\{1,j\}\(u\)^\{2\}\-H\_\{2,j\}\(u\)\\right\)\\leq Q\_\{j\}\(u\)H\_\{1,j\}\(u\)^\{2\}\.
Foru∈\[a,1\]u\\in\[a,1\], wherea\>0a\>0,Qj​\(u\)≤e​j−u≤e​j−a,Q\_\{j\}\(u\)\\leq e\\,j^\{\-u\}\\leq e\\,j^\{\-a\},andH1,j​\(u\)≤1\+log⁡j,H\_\{1,j\}\(u\)\\leq 1\+\\log j,

Consequently,

\|∂u2ψ⁡\(u,j\)\|\\displaystyle\\left\|\\partial\_\{u\}^\{2\}\\psi\(u,j\)\\right\|≤Qj​\(u\)j​\[2\+2​H1,j​\(u\)\+14​H1,j​\(u\)2\]\\displaystyle\\leq\\frac\{Q\_\{j\}\(u\)\}\{j\}\\Bigl\[2\+2H\_\{1,j\}\(u\)\+\\frac\{1\}\{4\}H\_\{1,j\}\(u\)^\{2\}\\Bigr\]≤ej1\+a​\[2\+2​\(1\+log⁡j\)\+14​\(1\+log⁡j\)2\]\.\\displaystyle\\leq\\frac\{e\}\{j^\{1\+a\}\}\\left\[2\+2\(1\+\\log j\)\+\\frac\{1\}\{4\}\(1\+\\log j\)^\{2\}\\right\]\.By simplification, we get

\|∂u2ψ⁡\(u,j\)\|≤94​e​\(1\+log⁡j\)2​j−\(1\+a\)\\left\|\\partial\_\{u\}^\{2\}\\psi\(u,j\)\\right\|\\leq\\frac\{9\}\{4\}e\(1\+\\log j\)^\{2\}j^\{\-\(1\+a\)\}for everyu∈\[a,1\]u\\in\[a,1\]\. Hence

S2​\(a\)≤94​e​\[ζ⁡\(1\+a\)−2​ζ′​\(1\+a\)\+ζ′′​\(1\+a\)\]\.S\_\{2\}\(a\)\\leq\\frac\{9\}\{4\}e\\left\[\\zeta\(1\+a\)\-2\\zeta^\{\\prime\}\(1\+a\)\+\\zeta^\{\\prime\\prime\}\(1\+a\)\\right\]\.\(33\)For coordinateii, simply chooseα¯i,loc=αi,⋆2\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\}=\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\. Then the result holds for allα∈\[α¯i,loc,1\]\\alpha\\in\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\]\. Using𝔼​\|xs\(i\)\|2≤σ2​∑m=0∞‖ei⊤​Gm‖22≤σ2​C~G2​ζ​\(2\)\\mathbb\{E\}\|x\_\{s\}^\{\(i\)\}\|^\{2\}\\leq\\sigma^\{2\}\\sum\_\{m=0\}^\{\\infty\}\\\|e\_\{i\}^\{\\top\}G\_\{m\}\\\|\_\{2\}^\{2\}\\leq\\sigma^\{2\}\\tilde\{C\}\_\{G\}^\{2\}\\zeta\(2\)and Minkowski’s inequality

‖ρs\(i\)​\(α\)‖L2\\displaystyle\\left\\\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\)\\right\\\|\_\{L^\{2\}\}≤\(α−αi,⋆\)2​∫01\(1−τ\)​∑j=1s\+1\|∂α2ψ⁡\(αi,∗\+τ⁡\(α−αi,⋆\),j\)\|​‖xs\+1−j\(i\)‖L2​𝑑τ\\displaystyle\\leq\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{2\}\\int\_\{0\}^\{1\}\(1\-\\tau\)\\sum\_\{j=1\}^\{s\+1\}\\left\|\\partial\_\{\\alpha\}^\{2\}\\psi\(\\alpha\_\{i,\*\}\+\\tau\(\\alpha\-\\alpha\_\{i,\\star\}\),j\)\\right\|\\left\\\|x\_\{s\+1\-j\}^\{\(i\)\}\\right\\\|\_\{L^\{2\}\}\\,d\\tau≤\(α−αi,⋆\)22​S2​\(a\)​σ2​C~G2​ζ​\(2\)\.\\displaystyle\\leq\\frac\{\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{2\}\}\{2\}S\_\{2\}\(a\)\\sqrt\{\\sigma^\{2\}\\tilde\{C\}\_\{G\}^\{2\}\\zeta\(2\)\}\.Squaring, we have

𝔼​\|ρs\(i\)​\(α\)\|2≤\(α−αi,⋆\)44​S2​\(αi,⋆2\)2​σ2​C~G2​ζ​\(2\)\\mathbb\{E\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\)\|^\{2\}\\leq\\frac\{\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{4\}\}\{4\}S\_\{2\}\(\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\)^\{2\}\\sigma^\{2\}\\tilde\{C\}\_\{G\}^\{2\}\\zeta\(2\)Hence, we have

1t​∑s=0t−1𝔼​\|bs\(i\)​\(α\)−\(α−αi,⋆\)​∑j≥1∂αψ⁡\(αi,⋆,j\)​xs\+1−j\(i\)\|2≤Ki24​\(α−αi,⋆\)4\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\)\-\(\\alpha\-\\alpha\_\{i,\\star\}\)\\sum\_\{j\\geq 1\}\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\)x\_\{s\+1\-j\}^\{\(i\)\}\\right\|^\{2\}\\leq\\frac\{K\_\{i\}^\{2\}\}\{4\}\(\\alpha\-\\alpha\_\{i,\\star\}\)^\{4\}withKi=94​e​σ​C~G​ζ⁡\(2\)​\[ζ⁡\(1\+αi,⋆2\)−2​ζ′​\(1\+αi,⋆2\)\+ζ′′​\(1\+αi,⋆2\)\]K\_\{i\}=\\frac\{9\}\{4\}e\\sigma\\tilde\{C\}\_\{G\}\\sqrt\{\\zeta\(2\)\}\\left\[\\zeta\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\)\-2\\zeta^\{\\prime\}\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\)\+\\zeta^\{\\prime\\prime\}\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\)\\right\]∎

Recall the row\-wise population risk

Rt\(i\)​\(αi\):=1t​infai∑s=0t−1𝔼⁡\[\(Δαi​xs\+1\(i\)−ai​xs\)2\]\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\frac\{1\}\{t\}\\inf\_\{a\_\{i\}\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\\left\(\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}\\right\)^\{2\}\\right\]\.Using the true dynamics,

Δαi​xs\+1\(i\)−ai​xs=ηs\(i\)\+bs\(i\)​\(αi\)−Δ​ai​xs\.\\Delta^\{\\alpha\_\{i\}\}x\_\{s\+1\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}=\\eta\_\{s\}^\{\(i\)\}\+b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\.Sincebs\(i\)​\(αi\)b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)andxsx\_\{s\}areℱs\\mathcal\{F\}\_\{s\}\-measurable and𝔼⁡\[ηs\(i\)∣ℱs\]=0\\mathbb\{E\}\[\\eta\_\{s\}^\{\(i\)\}\\mid\\mathcal\{F\}\_\{s\}\]=0,

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)=1t​infΔ​ai∑s=0t−1𝔼⁡\[\|bs\(i\)​\(αi\)−Δ​ai​xs\|2\]\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)=\\frac\{1\}\{t\}\\inf\_\{\\Delta a\_\{i\}\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\\left\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-\\Delta a\_\{i\}x\_\{s\}\\right\|^\{2\}\\right\]\.
Then we use the lower isometry argument in two stages\. First, a global profiled lower isometry bound in[Lemma12](https://arxiv.org/html/2609.18127#Thmlemma12)shows that, with high probability, simultaneously over all rows and all grid points outside the separation neighborhood ofαi,⋆\\alpha\_\{i,\\star\},

𝒬¯t,i​\(αi\)≥t2​\(Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\)\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\}\{2\}\\left\(R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\)\.Since each𝒜ϵ,i\\mathcal\{A\}\_\{\\epsilon,i\}is finite, the uniform statement is obtained by establishing the bound for a fixedαi\\alpha\_\{i\}and taking a union bound over the grid points and rows\. Combined with the in\-sample upper bound and the population separation gapγ\\gamma, this global bound rules out grid points outside the separation neighborhood and localizes the estimator to the set

𝒢i:=\{αi∈𝒜ϵ,i∩\[αi,loc,1\]:\|αi−αi,⋆\|≤μt,i2​Ki\}\.\\mathcal\{G\}\_\{i\}:=\\left\\\{\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\alpha\_\{i,\\mathrm\{loc\}\},1\]:\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\leq\\frac\{\\sqrt\{\\mu\_\{t,i\}\}\}\{2K\_\{i\}\}\\right\\\}\.\(34\)Within𝒢i\\mathcal\{G\}\_\{i\}, the population risk has a quadratic local curvature\. In particular, by Lemma[9](https://arxiv.org/html/2609.18127#Thmlemma9)and the reverse triangle inequality,

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)≥\|αi−αi,⋆\|​μt,i−Ki2​\|αi−αi,⋆\|2\.\\sqrt\{R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\}\\geq\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\sqrt\{\\mu\_\{t,i\}\}\-\\frac\{K\_\{i\}\}\{2\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.Hence, whenever\|αi−αi,⋆\|≤μt,i/\(2​Ki\)\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\leq\\sqrt\{\\mu\_\{t,i\}\}/\(2K\_\{i\}\),

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)≥μt,i2​\|αi−αi,⋆\|2\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\geq\\frac\{\\mu\_\{t,i\}\}\{2\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.The corresponding local lower isometry argument in[Lemma2](https://arxiv.org/html/2609.18127#Thmlemma2)transfers this curvature to the empirical profiled error and gives, uniformly overαi∈𝒢i\\alpha\_\{i\}\\in\\mathcal\{G\}\_\{i\},

𝒬¯t,i​\(αi\)≥t​μt,i8​\|αi−αi,⋆\|2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.Therefore, onceα^i∈𝒢i\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\}for everyii,

𝒬t,i​\(α^i,a^i−ai,⋆\)≥𝒬¯t,i​\(α^i\)≥t​μt,i8​\|α^i−αi,⋆\|2\.\\mathcal\{Q\}\_\{t,i\}\\left\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\right\)\\geq\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.Summing over the rows yields

ℰt≥t8​∑i=1nμt,i​\|α^i−αi,⋆\|2≥t​μmin8​‖𝜶^−𝜶⋆‖∞2\.\\mathcal\{E\}\_\{t\}\\geq\\frac\{t\}\{8\}\\sum\_\{i=1\}^\{n\}\\mu\_\{t,i\}\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\\geq\\frac\{t\\mu\_\{\\min\}\}\{8\}\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\.Combining this lower bound with the in\-sample upper bound and choosing the tunable coefficient sufficiently small allows the quadratic estimation\-error term on the upper\-bound side to be absorbed\. This gives

‖𝜶^−𝜶⋆‖∞2≲ϵmax2\+𝒪⁡\(t−1\)\.\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\lesssim\\epsilon\_\{\\max\}^\{2\}\+\\mathcal\{O\}\(t^\{\-1\}\)\.
Common notation for Lemmas[10](https://arxiv.org/html/2609.18127#Thmlemma10)–[13](https://arxiv.org/html/2609.18127#Thmlemma13)\.For each rowi∈\[n\]i\\in\[n\]and candidateαi\\alpha\_\{i\}, let

hi​\(αi\)\\displaystyle h\_\{i\}\(\\alpha\_\{i\}\):=αi−αi,⋆,\\displaystyle:=\\alpha\_\{i\}\-\\alpha\_\{i,\\star\},\(35\)bs\(i\)​\(αi\)\\displaystyle b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\(Δαi−Δαi,⋆\)​xs\+1\(i\),\\displaystyle:=\\bigl\(\\Delta^\{\\alpha\_\{i\}\}\-\\Delta^\{\\alpha\_\{i,\\star\}\}\\bigr\)x\_\{s\+1\}^\{\(i\)\},\(36\)gs\(i\)\\displaystyle g\_\{s\}^\{\(i\)\}:=∑j≥1di,j​xs\+1−j\(i\),di,j:=∂αψ⁡\(αi,⋆,j\),\\displaystyle:=\\sum\_\{j\\geq 1\}d\_\{i,j\}x\_\{s\+1\-j\}^\{\(i\)\},\\qquad d\_\{i,j\}:=\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\),\(37\)ρs\(i\)​\(αi\)\\displaystyle\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\):=bs\(i\)​\(αi\)−hi​\(αi\)​gs\(i\)\.\\displaystyle:=b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-h\_\{i\}\(\\alpha\_\{i\}\)g\_\{s\}^\{\(i\)\}\.\(38\)Recall the unprofiled and profiled noiseless errors

𝒬t,i​\(αi,v\)\\displaystyle\\mathcal\{Q\}\_\{t,i\}\(\\alpha\_\{i\},v\):=∑s=0t−1\|bs\(i\)​\(αi\)−v​xs\|2,\\displaystyle:=\\sum\_\{s=0\}^\{t\-1\}\\bigl\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-vx\_\{s\}\\bigr\|^\{2\},\(39\)𝒬¯t,i​\(αi\)\\displaystyle\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\):=infv∈ℝ1×n𝒬t,i​\(αi,v\)\.\\displaystyle:=\\inf\_\{v\\in\\mathbb\{R\}^\{1\\times n\}\}\\mathcal\{Q\}\_\{t,i\}\(\\alpha\_\{i\},v\)\.\(40\)Define the empirical and population profiled derivative curvatures using the same profiling operation:

μ^t,i\\displaystyle\\hat\{\\mu\}\_\{t,i\}:=infv∈ℝ1×n1t​∑s=0t−1\(gs\(i\)−v​xs\)2,\\displaystyle:=\\inf\_\{v\\in\\mathbb\{R\}^\{1\\times n\}\}\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\bigl\(g\_\{s\}^\{\(i\)\}\-vx\_\{s\}\\bigr\)^\{2\},\(41\)μt,i\\displaystyle\\mu\_\{t,i\}:=infv∈ℝ1×n1t​∑s=0t−1𝔼​\(gs\(i\)−v​xs\)2\.\\displaystyle:=\\inf\_\{v\\in\\mathbb\{R\}^\{1\\times n\}\}\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\bigl\(g\_\{s\}^\{\(i\)\}\-vx\_\{s\}\\bigr\)^\{2\}\.\(42\)Define

ri,sep:=μi,lb2​Ki\.r\_\{i,\\mathrm\{sep\}\}:=\\frac\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\{2K\_\{i\}\}\.\(43\)
###### Lemma 10\(Row\-wise lower isometry\)\.

Suppose that the following two events hold simultaneously:

μ^t,i\\displaystyle\\hat\{\\mu\}\_\{t,i\}≥12​μt,i,\\displaystyle\\geq\\frac\{1\}\{2\}\\mu\_\{t,i\},i∈\[n\],\\displaystyle i\\in\[n\],\(E1\)1t​∑s=0t−1\|ρs\(i\)​\(αi\)\|2\\displaystyle\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}≤Ki22​\|αi−αi,⋆\|4,\\displaystyle\\leq\\frac\{K\_\{i\}^\{2\}\}\{2\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{4\},i∈\[n\],αi∈𝒜ϵ,i∩\[α¯i,loc,1\]\.\\displaystyle i\\in\[n\],\\quad\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\]\.\(E2\)Then, simultaneously for everyi∈\[n\]i\\in\[n\]and everyαi∈𝒢i\\alpha\_\{i\}\\in\\mathcal\{G\}\_\{i\},

𝒬¯t,i​\(αi\)≥t​μt,i8​\|αi−αi,⋆\|2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.\(44\)Consequently, on any event on whichα^i∈𝒢i\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\},

𝒬t,i​\(α^i,a^i−ai,⋆\)≥t​μt,i8​\|α^i−αi,⋆\|2\.\\mathcal\{Q\}\_\{t,i\}\\bigl\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\bigr\)\\geq\\frac\{t\\mu\_\{t,i\}\}\{8\}\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}\.\(45\)

###### Proof\.

Fixiiandαi∈𝒢i\\alpha\_\{i\}\\in\\mathcal\{G\}\_\{i\}, and writehi:=αi−αi,⋆h\_\{i\}:=\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\. Let

Xt=\[x0,…,xt−1\],PX:=Xt⊤​\(Xt​Xt⊤\)†​Xt,X\_\{t\}=\[x\_\{0\},\\ldots,x\_\{t\-1\}\],\\qquad P\_\{X\}:=X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\\dagger\}X\_\{t\},where†\\daggerdenotes the Moore–Penrose pseudoinverse\. Define the corresponding time\-stacked vectors

bi​\(αi\)\\displaystyle b\_\{i\}\(\\alpha\_\{i\}\):=\(b0\(i\)​\(αi\),…,bt−1\(i\)​\(αi\)\)⊤,\\displaystyle:=\\bigl\(b\_\{0\}^\{\(i\)\}\(\\alpha\_\{i\}\),\\ldots,b\_\{t\-1\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\bigr\)^\{\\top\},gi\\displaystyle g\_\{i\}:=\(g0\(i\),…,gt−1\(i\)\)⊤,\\displaystyle:=\\bigl\(g\_\{0\}^\{\(i\)\},\\ldots,g\_\{t\-1\}^\{\(i\)\}\\bigr\)^\{\\top\},ρi​\(αi\)\\displaystyle\\rho\_\{i\}\(\\alpha\_\{i\}\):=\(ρ0\(i\)​\(αi\),…,ρt−1\(i\)​\(αi\)\)⊤\.\\displaystyle:=\\bigl\(\\rho\_\{0\}^\{\(i\)\}\(\\alpha\_\{i\}\),\\ldots,\\rho\_\{t\-1\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\bigr\)^\{\\top\}\.ThenPXP\_\{X\}is the orthogonal projection onto the row space ofXtX\_\{t\}, and profiling gives

𝒬¯t,i​\(αi\)=‖\(I−PX\)​bi​\(αi\)‖22\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)=\\\|\(I\-P\_\{X\}\)b\_\{i\}\(\\alpha\_\{i\}\)\\\|\_\{2\}^\{2\}\.Sincebi​\(αi\)=hi​gi\+ρi​\(αi\)b\_\{i\}\(\\alpha\_\{i\}\)=h\_\{i\}g\_\{i\}\+\\rho\_\{i\}\(\\alpha\_\{i\}\), the reverse triangle inequality and‖I−PX‖o​p≤1\\\|I\-P\_\{X\}\\\|\_\{op\}\\leq 1imply

𝒬¯t,i​\(αi\)t\\displaystyle\\sqrt\{\\frac\{\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\}\{t\}\}≥\|hi\|​μ^t,i−\(1t​∑s=0t−1\|ρs\(i\)​\(αi\)\|2\)1/2\.\\displaystyle\\geq\|h\_\{i\}\|\\sqrt\{\\hat\{\\mu\}\_\{t,i\}\}\-\\left\(\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\\right\)^\{1/2\}\.On \([E1](https://arxiv.org/html/2609.18127#A2.Ex109)\)–\([E2](https://arxiv.org/html/2609.18127#A2.Ex110)\),

𝒬¯t,i​\(αi\)t≥\|hi\|​μt,i2−Ki2​\|hi\|2\.\\sqrt\{\\frac\{\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\}\{t\}\}\\geq\|h\_\{i\}\|\\sqrt\{\\frac\{\\mu\_\{t,i\}\}\{2\}\}\-\\frac\{K\_\{i\}\}\{\\sqrt\{2\}\}\|h\_\{i\}\|^\{2\}\.Becauseαi∈𝒢i\\alpha\_\{i\}\\in\\mathcal\{G\}\_\{i\},Ki​\|hi\|≤μt,i/2K\_\{i\}\|h\_\{i\}\|\\leq\\sqrt\{\\mu\_\{t,i\}\}/2, and hence

𝒬¯t,i​\(αi\)t≥μt,i2​2​\|hi\|\.\\sqrt\{\\frac\{\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\}\{t\}\}\\geq\\frac\{\\sqrt\{\\mu\_\{t,i\}\}\}\{2\\sqrt\{2\}\}\|h\_\{i\}\|\.Squaring proves \([44](https://arxiv.org/html/2609.18127#A2.E44)\)\. Finally,

𝒬t,i​\(α^i,a^i−ai,⋆\)≥𝒬¯t,i​\(α^i\),\\mathcal\{Q\}\_\{t,i\}\\bigl\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\bigr\)\\geq\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\),which proves \([45](https://arxiv.org/html/2609.18127#A2.E45)\) wheneverα^i∈𝒢i\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\}\. ∎

###### Lemma 11\(Relative concentration of the empirical derivative curvature\)\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds\. Let

Di:=∑j≥1\|di,j\|,D2,i:=\(∑j≥1di,j2\)1/2,ℒx:=C~x​\(1\+2αmin\)\.D\_\{i\}:=\\sum\_\{j\\geq 1\}\|d\_\{i,j\}\|,\\qquad D\_\{2,i\}:=\\left\(\\sum\_\{j\\geq 1\}d\_\{i,j\}^\{2\}\\right\)^\{1/2\},\\qquad\\mathcal\{L\}\_\{x\}:=\\widetilde\{C\}\_\{x\}\\left\(1\+\\frac\{2\}\{\\alpha\_\{\\min\}\}\\right\)\.Define

zs\(i\):=\[xsgs\(i\)\],Σ^z,i:=1t​∑s=0t−1zs\(i\)​zs\(i\)⊤,Σz,i:=𝔼​Σ^z,i\.z\_\{s\}^\{\(i\)\}:=\\begin\{bmatrix\}x\_\{s\}\\\\ g\_\{s\}^\{\(i\)\}\\end\{bmatrix\},\\qquad\\hat\{\\Sigma\}\_\{z,i\}:=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}z\_\{s\}^\{\(i\)\}z\_\{s\}^\{\(i\)\\top\},\\qquad\\Sigma\_\{z,i\}:=\\mathbb\{E\}\\hat\{\\Sigma\}\_\{z,i\}\.Set

κi,⋆:=di,22\+di,32,\\kappa\_\{i,\\star\}:=d\_\{i,2\}^\{2\}\+d\_\{i,3\}^\{2\},Cder:=max⁡\{2​‖G1‖op2\+3​\(‖G2‖op\+‖G1‖op2\)2,2\+3​‖G1‖op2,3\},C\_\{\\mathrm\{der\}\}:=\\max\\left\\\{2\\\|G\_\{1\}\\\|\_\{\\mathrm\{op\}\}^\{2\}\+3\\bigl\(\\\|G\_\{2\}\\\|\_\{\\mathrm\{op\}\}\+\\\|G\_\{1\}\\\|\_\{\\mathrm\{op\}\}^\{2\}\\bigr\)^\{2\},\\;2\+3\\\|G\_\{1\}\\\|\_\{\\mathrm\{op\}\}^\{2\},\\;3\\right\\\},and

μi,lb:=σ2​κi,⋆2​Cder\.\\mu\_\{i,\\mathrm\{lb\}\}:=\\frac\{\\sigma^\{2\}\\kappa\_\{i,\\star\}\}\{2C\_\{\\mathrm\{der\}\}\}\.For eachii, define

ℜ¯i:=ℒx​\[2σ\+Di\+2​ℒxσ​D2,iμi,lb\]2,\\overline\{\\mathfrak\{R\}\}\_\{i\}:=\\mathcal\{L\}\_\{x\}\\left\[\\frac\{\\sqrt\{2\}\}\{\\sigma\}\+\\frac\{D\_\{i\}\+\\frac\{\\sqrt\{2\\mathcal\{L\}\_\{x\}\}\}\{\\sigma\}D\_\{2,i\}\}\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\\right\]^\{2\},and let

q1​\(δ\):=\(n\+1\)​log⁡9\+log⁡4​nδ\.q\_\{1\}\(\\delta\):=\(n\+1\)\\log 9\+\\log\\frac\{4n\}\{\\delta\}\.
Then, fort≥6t\\geq 6,Σz,i≻0\\Sigma\_\{z,i\}\\succ 0andμt,i≥μi,lb\>0\\mu\_\{t,i\}\\geq\\mu\_\{i,\\mathrm\{lb\}\}\>0\. Moreover, with probability at least1−δ/21\-\\delta/2, simultaneously for alli∈\[n\]i\\in\[n\],

‖Σz,i−1/2\(Σ^z,i−Σz,i\)Σz,i−1/2‖op≤4\(ℜ¯i​q1​\(δ\)t\+ℜ¯i​q1​\(δ\)t\)\.\\left\\\|\\Sigma\_\{z,i\}^\{\-1/2\}\\bigl\(\\hat\{\\Sigma\}\_\{z,i\}\-\\Sigma\_\{z,i\}\\bigr\)\\Sigma\_\{z,i\}^\{\-1/2\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq 4\\left\(\\sqrt\{\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\}\+\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\\right\)\.
Consequently, if

t≥tE1​\(δ\):=256​q1​\(δ\)​maxi∈\[n\]​ℜ¯i,t\\geq t\_\{\\mathrm\{E1\}\}\(\\delta\):=256q\_\{1\}\(\\delta\)\\max\_\{i\\in\[n\]\}\\overline\{\\mathfrak\{R\}\}\_\{i\},then, with probability at least1−δ/21\-\\delta/2,

Σ^z,i⪰12​Σz,i,i∈\[n\],\\hat\{\\Sigma\}\_\{z,i\}\\succeq\\frac\{1\}\{2\}\\Sigma\_\{z,i\},\\qquad i\\in\[n\],and hence

μ^t,i≥12​μt,i,i∈\[n\]\.\\hat\{\\mu\}\_\{t,i\}\\geq\\frac\{1\}\{2\}\\mu\_\{t,i\},\\qquad i\\in\[n\]\.

###### Proof\.

For

hi,ℓ⊤:=∑j=1ℓdi,j​ei⊤​Gℓ−j,h\_\{i,\\ell\}^\{\\top\}:=\\sum\_\{j=1\}^\{\\ell\}d\_\{i,j\}e\_\{i\}^\{\\top\}G\_\{\\ell\-j\},independence of the innovations gives

1t​∑s=0t−1𝔼​\(gs\(i\)−v​xs\)2=σ2t​∑ℓ=1t−1\(t−ℓ\)​‖hi,ℓ⊤−v​Gℓ−1‖22\.\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\bigl\(g\_\{s\}^\{\(i\)\}\-vx\_\{s\}\\bigr\)^\{2\}=\\frac\{\\sigma^\{2\}\}\{t\}\\sum\_\{\\ell=1\}^\{t\-1\}\(t\-\\ell\)\\\|h\_\{i,\\ell\}^\{\\top\}\-vG\_\{\\ell\-1\}\\\|\_\{2\}^\{2\}\.Fort≥6t\\geq 6, retaining onlyℓ=1,2,3\\ell=1,2,3yields

μt,i≥σ22​infv∑ℓ=13‖hi,ℓ⊤−v​Gℓ−1‖22\.\\mu\_\{t,i\}\\geq\\frac\{\\sigma^\{2\}\}\{2\}\\inf\_\{v\}\\sum\_\{\\ell=1\}^\{3\}\\\|h\_\{i,\\ell\}^\{\\top\}\-vG\_\{\\ell\-1\}\\\|\_\{2\}^\{2\}\.If

r=di,1​ei⊤−v,r=d\_\{i,1\}e\_\{i\}^\{\\top\}\-v,u2=r​G1\+di,2​ei⊤,u3=r​G2\+di,2​ei⊤​G1\+di,3​ei⊤,u\_\{2\}=rG\_\{1\}\+d\_\{i,2\}e\_\{i\}^\{\\top\},\\qquad u\_\{3\}=rG\_\{2\}\+d\_\{i,2\}e\_\{i\}^\{\\top\}G\_\{1\}\+d\_\{i,3\}e\_\{i\}^\{\\top\},then

κi,⋆≤Cder​\(‖r‖22\+‖u2‖22\+‖u3‖22\)\.\\kappa\_\{i,\\star\}\\leq C\_\{\\mathrm\{der\}\}\\left\(\\\|r\\\|\_\{2\}^\{2\}\+\\\|u\_\{2\}\\\|\_\{2\}^\{2\}\+\\\|u\_\{3\}\\\|\_\{2\}^\{2\}\\right\)\.Therefore,

μt,i≥σ2​κi,⋆2​Cder=μi,lb\.\\mu\_\{t,i\}\\geq\\frac\{\\sigma^\{2\}\\kappa\_\{i,\\star\}\}\{2C\_\{\\mathrm\{der\}\}\}=\\mu\_\{i,\\mathrm\{lb\}\}\.
Next write

Σz,i=\[Σx,tct,ict,i⊤qt,i\],\\Sigma\_\{z,i\}=\\begin\{bmatrix\}\\Sigma\_\{x,t\}&c\_\{t,i\}\\\\ c\_\{t,i\}^\{\\top\}&q\_\{t,i\}\\end\{bmatrix\},where

Σx,t:=1t​∑s=0t−1𝔼⁡\[xs​xs⊤\],qt,i:=1t​∑s=0t−1𝔼⁡\[\(gs\(i\)\)2\]\.\\Sigma\_\{x,t\}:=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\[x\_\{s\}x\_\{s\}^\{\\top\}\],\\qquad q\_\{t,i\}:=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\bigl\[\(g\_\{s\}^\{\(i\)\}\)^\{2\}\\bigr\]\.Sincexsx\_\{s\}contains the fresh innovationηs−1\\eta\_\{s\-1\}fors≥1s\\geq 1,

Σx,t⪰σ22​In\.\\Sigma\_\{x,t\}\\succeq\\frac\{\\sigma^\{2\}\}\{2\}I\_\{n\}\.The Schur complement ofΣx,t\\Sigma\_\{x,t\}inΣz,i\\Sigma\_\{z,i\}isμt,i\>0\\mu\_\{t,i\}\>0\. HenceΣz,i≻0\\Sigma\_\{z,i\}\\succ 0\.

We now derive a deterministic bound on the temporal dependence of the whitened process\. Let

at,i:=Σx,t−1​ct,i\.a\_\{t,i\}:=\\Sigma\_\{x,t\}^\{\-1\}c\_\{t,i\}\.The Schur\-complement identity gives

qt,i=μt,i\+at,i⊤​Σx,t​at,i\.q\_\{t,i\}=\\mu\_\{t,i\}\+a\_\{t,i\}^\{\\top\}\\Sigma\_\{x,t\}a\_\{t,i\}\.In particular,

at,i⊤​Σx,t​at,i≤qt,i\.a\_\{t,i\}^\{\\top\}\\Sigma\_\{x,t\}a\_\{t,i\}\\leq q\_\{t,i\}\.
We first boundqt,iq\_\{t,i\}\. The covariance decay in[Lemma14](https://arxiv.org/html/2609.18127#Thmlemma14)gives, for everyss,

𝔼⁡\[\(gs\(i\)\)2\]\\displaystyle\\mathbb\{E\}\\bigl\[\(g\_\{s\}^\{\(i\)\}\)^\{2\}\\bigr\]≤C~x​∑j,k≥1\|di,j\|​\|di,k\|​\(\|j−k\|\+1\)−\(1\+αmin\)\.\\displaystyle\\leq\\widetilde\{C\}\_\{x\}\\sum\_\{j,k\\geq 1\}\|d\_\{i,j\}\|\|d\_\{i,k\}\|\\bigl\(\|j\-k\|\+1\\bigr\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\.Let

γk:=\(\|k\|\+1\)−\(1\+αmin\),k∈ℤ\.\\gamma\_\{k\}:=\(\|k\|\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\},\\qquad k\\in\\mathbb\{Z\}\.Young’s convolution inequality gives

∑j,k≥1\|di,j\|​\|di,k\|​γj−k≤‖γ‖ℓ1​\(ℤ\)​∑j≥1di,j2\.\\sum\_\{j,k\\geq 1\}\|d\_\{i,j\}\|\|d\_\{i,k\}\|\\gamma\_\{j\-k\}\\leq\\\|\\gamma\\\|\_\{\\ell\_\{1\}\(\\mathbb\{Z\}\)\}\\sum\_\{j\\geq 1\}d\_\{i,j\}^\{2\}\.Moreover,

‖γ‖ℓ1​\(ℤ\)≤1\+2αmin\.\\\|\\gamma\\\|\_\{\\ell\_\{1\}\(\\mathbb\{Z\}\)\}\\leq 1\+\\frac\{2\}\{\\alpha\_\{\\min\}\}\.Consequently,

qt,i≤ℒx​D2,i2\.q\_\{t,i\}\\leq\\mathcal\{L\}\_\{x\}D\_\{2,i\}^\{2\}\.SinceΣx,t⪰σ2​In/2\\Sigma\_\{x,t\}\\succeq\\sigma^\{2\}I\_\{n\}/2, it follows that

‖at,i‖22≤2​qt,iσ2≤2​ℒx​D2,i2σ2,\\\|a\_\{t,i\}\\\|\_\{2\}^\{2\}\\leq\\frac\{2q\_\{t,i\}\}\{\\sigma^\{2\}\}\\leq\\frac\{2\\mathcal\{L\}\_\{x\}D\_\{2,i\}^\{2\}\}\{\\sigma^\{2\}\},and therefore

‖at,i‖2≤2​ℒxσ​D2,i\.\\\|a\_\{t,i\}\\\|\_\{2\}\\leq\\frac\{\\sqrt\{2\\mathcal\{L\}\_\{x\}\}\}\{\\sigma\}D\_\{2,i\}\.
Fixiiand a unit vectoru∈ℝn\+1u\\in\\mathbb\{R\}^\{n\+1\}, and write

w:=Σz,i−1/2u=\[βb\]\.w:=\\Sigma\_\{z,i\}^\{\-1/2\}u=\\begin\{bmatrix\}\\beta\\\\ b\\end\{bmatrix\}\.Sincew⊤​Σz,i​w=1w^\{\\top\}\\Sigma\_\{z,i\}w=1, the factorization

Σz,i=\[Iat,i01\]⊤​\[Σx,t00μt,i\]​\[Iat,i01\]\\Sigma\_\{z,i\}=\\begin\{bmatrix\}I&a\_\{t,i\}\\\\ 0&1\\end\{bmatrix\}^\{\\\!\\top\}\\begin\{bmatrix\}\\Sigma\_\{x,t\}&0\\\\ 0&\\mu\_\{t,i\}\\end\{bmatrix\}\\begin\{bmatrix\}I&a\_\{t,i\}\\\\ 0&1\\end\{bmatrix\}implies

1=\(β\+b​at,i\)⊤​Σx,t​\(β\+b​at,i\)\+μt,i​b2\.1=\(\\beta\+ba\_\{t,i\}\)^\{\\top\}\\Sigma\_\{x,t\}\(\\beta\+ba\_\{t,i\}\)\+\\mu\_\{t,i\}b^\{2\}\.Therefore,

‖β\+b​at,i‖2≤2σ,\|b\|≤1μi,lb\.\\\|\\beta\+ba\_\{t,i\}\\\|\_\{2\}\\leq\\frac\{\\sqrt\{2\}\}\{\\sigma\},\\qquad\|b\|\\leq\\frac\{1\}\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\.It follows that

‖β‖2\+Di​\|b\|\\displaystyle\\\|\\beta\\\|\_\{2\}\+D\_\{i\}\|b\|≤‖β\+b​at,i‖2\+\(‖at,i‖2\+Di\)​\|b\|\\displaystyle\\leq\\\|\\beta\+ba\_\{t,i\}\\\|\_\{2\}\+\\bigl\(\\\|a\_\{t,i\}\\\|\_\{2\}\+D\_\{i\}\\bigr\)\|b\|≤2σ\+Di\+2​ℒxσ​D2,iμi,lb\.\\displaystyle\\leq\\frac\{\\sqrt\{2\}\}\{\\sigma\}\+\\frac\{D\_\{i\}\+\\frac\{\\sqrt\{2\\mathcal\{L\}\_\{x\}\}\}\{\\sigma\}D\_\{2,i\}\}\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\.
Now define the scalar whitened process

rs\(i,u\):=u⊤Σz,i−1/2zs\(i\)=β⊤xs\+bgs\(i\)\.r\_\{s\}^\{\(i,u\)\}:=u^\{\\top\}\\Sigma\_\{z,i\}^\{\-1/2\}z\_\{s\}^\{\(i\)\}=\\beta^\{\\top\}x\_\{s\}\+bg\_\{s\}^\{\(i\)\}\.It is a linear filter ofxsx\_\{s\}whose coefficientℓ1\\ell\_\{1\}norm is at most

Fi:=2σ\+Di\+2​ℒxσ​D2,iμi,lb\.F\_\{i\}:=\\frac\{\\sqrt\{2\}\}\{\\sigma\}\+\\frac\{D\_\{i\}\+\\frac\{\\sqrt\{2\\mathcal\{L\}\_\{x\}\}\}\{\\sigma\}D\_\{2,i\}\}\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\.The covariance decay convolution bound in[Lemma14](https://arxiv.org/html/2609.18127#Thmlemma14)therefore gives

supp∑k∈ℤ\|Cov⁡\(rp\(i,u\),rp\+k\(i,u\)\)\|≤Fi2​ℒx=ℜ¯i\.\\sup\_\{p\}\\sum\_\{k\\in\\mathbb\{Z\}\}\\left\|\\operatorname\{Cov\}\\bigl\(r\_\{p\}^\{\(i,u\)\},r\_\{p\+k\}^\{\(i,u\)\}\\bigr\)\\right\|\\leq F\_\{i\}^\{2\}\\mathcal\{L\}\_\{x\}=\\overline\{\\mathfrak\{R\}\}\_\{i\}\.If

𝒯i,u:=Cov⁡\(r0\(i,u\),…,rt−1\(i,u\)\),\\mathcal\{T\}\_\{i,u\}:=\\operatorname\{Cov\}\\left\(r\_\{0\}^\{\(i,u\)\},\\ldots,r\_\{t\-1\}^\{\(i,u\)\}\\right\),then

‖𝒯i,u‖op≤ℜ¯i\.\\\|\\mathcal\{T\}\_\{i,u\}\\\|\_\{\\mathrm\{op\}\}\\leq\\overline\{\\mathfrak\{R\}\}\_\{i\}\.
Furthermore,

tr⁡\(𝒯i,u\)\\displaystyle\\operatorname\{tr\}\(\\mathcal\{T\}\_\{i,u\}\)=∑s=0t−1𝔼⁡\[\(rs\(i,u\)\)2\]\\displaystyle=\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\\bigl\(r\_\{s\}^\{\(i,u\)\}\\bigr\)^\{2\}\\right\]=tu⊤Σz,i−1/2Σz,iΣz,i−1/2u\\displaystyle=t\\,u^\{\\top\}\\Sigma\_\{z,i\}^\{\-1/2\}\\Sigma\_\{z,i\}\\Sigma\_\{z,i\}^\{\-1/2\}u=t\.\\displaystyle=t\.Since𝒯i,u⪰0\\mathcal\{T\}\_\{i,u\}\\succeq 0,

‖𝒯i,u‖F2≤‖𝒯i,u‖op​tr⁡\(𝒯i,u\)≤t​ℜ¯i\.\\\|\\mathcal\{T\}\_\{i,u\}\\\|\_\{F\}^\{2\}\\leq\\\|\\mathcal\{T\}\_\{i,u\}\\\|\_\{\\mathrm\{op\}\}\\operatorname\{tr\}\(\\mathcal\{T\}\_\{i,u\}\)\\leq t\\overline\{\\mathfrak\{R\}\}\_\{i\}\.
Let

r\(i,u\):=\(r0\(i,u\),…,rt−1\(i,u\)\)⊤\.r^\{\(i,u\)\}:=\\left\(r\_\{0\}^\{\(i,u\)\},\\ldots,r\_\{t\-1\}^\{\(i,u\)\}\\right\)^\{\\top\}\.The Gaussian quadratic\-form inequality\[hsu2012tail\]gives, for everyx\>0x\>0,

\|1t​‖r\(i,u\)‖22−1\|≤2​ℜ¯i​xt\+2​ℜ¯i​xt\\left\|\\frac\{1\}\{t\}\\\|r^\{\(i,u\)\}\\\|\_\{2\}^\{2\}\-1\\right\|\\leq 2\\sqrt\{\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}x\}\{t\}\}\+2\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}x\}\{t\}with probability at least1−2​e−x1\-2e^\{\-x\}\.

Equivalently,

\|u⊤Σz,i−1/2\(Σ^z,i−Σz,i\)Σz,i−1/2u\|≤2ℜ¯i​xt\+2ℜ¯i​xt\.\\left\|u^\{\\top\}\\Sigma\_\{z,i\}^\{\-1/2\}\\bigl\(\\hat\{\\Sigma\}\_\{z,i\}\-\\Sigma\_\{z,i\}\\bigr\)\\Sigma\_\{z,i\}^\{\-1/2\}u\\right\|\\leq 2\\sqrt\{\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}x\}\{t\}\}\+2\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}x\}\{t\}\.
Apply this inequality to a1/41/4\-net of the unit sphere inℝn\+1\\mathbb\{R\}^\{n\+1\}, whose cardinality is at most9n\+19^\{n\+1\}, and take a union bound overi∈\[n\]i\\in\[n\]\. With

x=q1​\(δ\)=\(n\+1\)​log⁡9\+log⁡4​nδ,x=q\_\{1\}\(\\delta\)=\(n\+1\)\\log 9\+\\log\\frac\{4n\}\{\\delta\},the total failure probability is at most

2​n​9n\+1​e−q1​\(δ\)=δ2\.2n9^\{n\+1\}e^\{\-q\_\{1\}\(\\delta\)\}=\\frac\{\\delta\}\{2\}\.The standard1/41/4\-net bound for symmetric matrices then yields

‖Σz,i−1/2\(Σ^z,i−Σz,i\)Σz,i−1/2‖op≤4\(ℜ¯i​q1​\(δ\)t\+ℜ¯i​q1​\(δ\)t\)\\left\\\|\\Sigma\_\{z,i\}^\{\-1/2\}\\bigl\(\\hat\{\\Sigma\}\_\{z,i\}\-\\Sigma\_\{z,i\}\\bigr\)\\Sigma\_\{z,i\}^\{\-1/2\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq 4\\left\(\\sqrt\{\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\}\+\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\\right\)simultaneously for allii\.

If

t≥256​q1​\(δ\)​maxi​ℜ¯i,t\\geq 256q\_\{1\}\(\\delta\)\\max\_\{i\}\\overline\{\\mathfrak\{R\}\}\_\{i\},then

ℜ¯i​q1​\(δ\)t≤116,ℜ¯i​q1​\(δ\)t≤1256,\\sqrt\{\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\}\\leq\\frac\{1\}\{16\},\\qquad\\frac\{\\overline\{\\mathfrak\{R\}\}\_\{i\}q\_\{1\}\(\\delta\)\}\{t\}\\leq\\frac\{1\}\{256\},so the preceding right\-hand side is at most

4​\(116\+1256\)=1764<12\.4\\left\(\\frac\{1\}\{16\}\+\\frac\{1\}\{256\}\\right\)=\\frac\{17\}\{64\}<\\frac\{1\}\{2\}\.Hence

Σz,i−1/2Σ^z,iΣz,i−1/2⪰12I,\\Sigma\_\{z,i\}^\{\-1/2\}\\hat\{\\Sigma\}\_\{z,i\}\\Sigma\_\{z,i\}^\{\-1/2\}\\succeq\\frac\{1\}\{2\}I,and therefore

Σ^z,i⪰12​Σz,i\.\\hat\{\\Sigma\}\_\{z,i\}\\succeq\\frac\{1\}\{2\}\\Sigma\_\{z,i\}\.
Finally, for

w⁡\(v\):=\(−v⊤,1\)⊤,w\(v\):=\(\-v^\{\\top\},1\)^\{\\top\},we have

μ^t,i=infvw​\(v\)⊤​Σ^z,i​w​\(v\)\\hat\{\\mu\}\_\{t,i\}=\\inf\_\{v\}w\(v\)^\{\\top\}\\hat\{\\Sigma\}\_\{z,i\}w\(v\)and

μt,i=infvw​\(v\)⊤​Σz,i​w​\(v\)\.\\mu\_\{t,i\}=\\inf\_\{v\}w\(v\)^\{\\top\}\\Sigma\_\{z,i\}w\(v\)\.Thus

μ^t,i≥12​μt,i,\\hat\{\\mu\}\_\{t,i\}\\geq\\frac\{1\}\{2\}\\mu\_\{t,i\},which proves event\(E1\)\\mathrm\{\(E1\)\}\. ∎

###### Lemma 12\(Global profiled lower isometry\)\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds\. Define

γ:=min⁡infαi∈𝒜ϵ,i\|hi​\(αi\)\|\>ri,sepi∈\[n\]⁡\{Rt\(i\)​\(αi\)−Rt\(i\)​\(αi∘\)\},\\gamma:=\\min\_\{i\\in\[n\]\}\\inf\_\{\\begin\{subarray\}\{c\}\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\\\ \|h\_\{i\}\(\\alpha\_\{i\}\)\|\>r\_\{i,\\mathrm\{sep\}\}\\end\{subarray\}\}\\left\\\{R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\\\},and suppose thatγ\>0\.\\gamma\>0\.LetFG:=2σ\+S1γ​\(1\+2​Lxσ\),F\_\{\\mathrm\{G\}\}:=\\frac\{\\sqrt\{2\}\}\{\\sigma\}\+\\frac\{S\_\{1\}\}\{\\sqrt\{\\gamma\}\}\\left\(1\+\\frac\{\\sqrt\{2L\_\{x\}\}\}\{\\sigma\}\\right\),and

qG​\(δ\):=\(n\+1\)​log⁡9\+log⁡2​∑i=1nMiδ\.q\_\{\\mathrm\{G\}\}\(\\delta\):=\(n\+1\)\\log 9\+\\log\\frac\{2\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\.Ift≥max⁡\{2,256​Lx​FG2​qG​\(δ\)\},t\\geq\\max\\left\\\{2,\\,256L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}q\_\{\\mathrm\{G\}\}\(\\delta\)\\right\\\},then with probability at least1−δ1\-\\delta, simultaneously for everyi∈\[n\]i\\in\[n\]and everyαi∈𝒜ϵ,i\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}satisfying\|hi​\(αi\)\|\>ri,sep\|h\_\{i\}\(\\alpha\_\{i\}\)\|\>r\_\{i,\\mathrm\{sep\}\},

𝒬¯t,i​\(αi\)≥t2​\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]≥t​γ2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\}\{2\}\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\\geq\\frac\{t\\gamma\}\{2\}\.

###### Proof\.

Fixi∈\[n\]i\\in\[n\]andαi∈𝒜ϵ,i\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}satisfying\|hi​\(αi\)\|\>ri,sep\|h\_\{i\}\(\\alpha\_\{i\}\)\|\>r\_\{i,\\mathrm\{sep\}\}\. Definezs\(i,αi\):=\[xsbs\(i\)​\(αi\)\],z\_\{s\}^\{\(i,\\alpha\_\{i\}\)\}:=\\begin\{bmatrix\}x\_\{s\}\\\\ b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\end\{bmatrix\},Σ^i,αi:=1t​∑s=0t−1zs\(i,αi\)​zs\(i,αi\)⊤,\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}:=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}z\_\{s\}^\{\(i,\\alpha\_\{i\}\)\}z\_\{s\}^\{\(i,\\alpha\_\{i\}\)\\top\},andΣi,αi:=𝔼​Σ^i,αi\.\\Sigma\_\{i,\\alpha\_\{i\}\}:=\\mathbb\{E\}\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}\.Forw⁡\(v\):=\[−v⊤1\],w\(v\):=\\begin\{bmatrix\}\-v^\{\\top\}\\\\ 1\\end\{bmatrix\},we have

w​\(v\)⊤​zs\(i,αi\)=bs\(i\)​\(αi\)−v​xs\.w\(v\)^\{\\top\}z\_\{s\}^\{\(i,\\alpha\_\{i\}\)\}=b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-vx\_\{s\}\.Hence, by the definition of the profiled noiseless error,

1t​𝒬¯t,i​\(αi\)=infvw​\(v\)⊤​Σ^i,αi​w​\(v\)\.\\frac\{1\}\{t\}\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)=\\inf\_\{v\}w\(v\)^\{\\top\}\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}w\(v\)\.\(46\)WriteΣi,αi=\[Σx,tci,αici,αi⊤qi,αi\],\\Sigma\_\{i,\\alpha\_\{i\}\}=\\begin\{bmatrix\}\\Sigma\_\{x,t\}&c\_\{i,\\alpha\_\{i\}\}\\\\ c\_\{i,\\alpha\_\{i\}\}^\{\\top\}&q\_\{i,\\alpha\_\{i\}\}\\end\{bmatrix\},whereci,αi=1t​∑s=0t−1𝔼⁡\[xs​bs\(i\)​\(αi\)\],c\_\{i,\\alpha\_\{i\}\}=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[x\_\{s\}b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\right\],andqi,αi=1t​∑s=0t−1𝔼⁡\[\|bs\(i\)​\(αi\)\|2\]\.q\_\{i,\\alpha\_\{i\}\}=\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\\right\]\.The population\-risk identity gives

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)=infv1t​∑s=0t−1𝔼⁡\[\|bs\(i\)​\(αi\)−v​xs\|2\]\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)=\\inf\_\{v\}\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\[\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-vx\_\{s\}\|^\{2\}\\right\]\.Expanding the square,

1t​∑s=0t−1𝔼​\|bs\(i\)​\(αi\)−v​xs\|2=qi,αi−2​v​ci,αi\+v​Σx,t​v⊤\.\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-vx\_\{s\}\|^\{2\}=q\_\{i,\\alpha\_\{i\}\}\-2vc\_\{i,\\alpha\_\{i\}\}\+v\\Sigma\_\{x,t\}v^\{\\top\}\.Fort≥2t\\geq 2, the innovation gives

Σx,t⪰σ22​In\.\\Sigma\_\{x,t\}\\succeq\\frac\{\\sigma^\{2\}\}\{2\}I\_\{n\}\.\(47\)ThusΣx,t≻0\\Sigma\_\{x,t\}\\succ 0, and minimizing the preceding quadratic function overvvyieldsv⋆=ci,αi⊤​Σx,t−1\.v^\{\\star\}=c\_\{i,\\alpha\_\{i\}\}^\{\\top\}\\Sigma\_\{x,t\}^\{\-1\}\.Consequently,

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)=qi,αi−ci,αi⊤​Σx,t−1​ci,αi\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)=q\_\{i,\\alpha\_\{i\}\}\-c\_\{i,\\alpha\_\{i\}\}^\{\\top\}\\Sigma\_\{x,t\}^\{\-1\}c\_\{i,\\alpha\_\{i\}\}\.\(48\)Equivalently,

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)=infvw​\(v\)⊤​Σi,αi​w​\(v\)\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)=\\inf\_\{v\}w\(v\)^\{\\top\}\\Sigma\_\{i,\\alpha\_\{i\}\}w\(v\)\.\(49\)
Since

Rt\(i\)​\(αi∘\)−Rt\(i\)​\(αi,⋆\)≥0,R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\geq 0,the definition ofγ\\gammaimplies

Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)≥γ\.R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\geq\\gamma\.\(50\)Hence the Schur complement in \([48](https://arxiv.org/html/2609.18127#A2.E48)\) is strictly positive andΣi,αi≻0\\Sigma\_\{i,\\alpha\_\{i\}\}\\succ 0\.

Let

Δ​ψj:=ψ⁡\(αi,j\)−ψ⁡\(αi,⋆,j\)\.\\Delta\\psi\_\{j\}:=\\psi\(\\alpha\_\{i\},j\)\-\\psi\(\\alpha\_\{i,\\star\},j\)\.Then

bs\(i\)​\(αi\)=∑j≥1Δ​ψj​xs\+1−j\(i\)\.b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)=\\sum\_\{j\\geq 1\}\\Delta\\psi\_\{j\}x\_\{s\+1\-j\}^\{\(i\)\}\.By the fundamental theorem of calculus and the definition ofS1S\_\{1\},

∑j≥1\|Δ​ψj\|≤S1​\|hi​\(αi\)\|\.\\sum\_\{j\\geq 1\}\|\\Delta\\psi\_\{j\}\|\\leq S\_\{1\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|\.\(51\)Therefore,

\(∑j≥1\|Δ​ψj\|2\)1/2≤S1​\|hi​\(αi\)\|\.\\left\(\\sum\_\{j\\geq 1\}\|\\Delta\\psi\_\{j\}\|^\{2\}\\right\)^\{1/2\}\\leq S\_\{1\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|\.\(52\)Letγk:=\(\|k\|\+1\)−\(1\+αmin\),\\gamma\_\{k\}:=\(\|k\|\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\},wherek∈ℤ\.k\\in\\mathbb\{Z\}\.The covariance\-decay bound in[Lemma14](https://arxiv.org/html/2609.18127#Thmlemma14)gives

\|Cov⁡\(xp\(i\),xq\(i\)\)\|≤C~x​γp−q\.\\left\|\\operatorname\{Cov\}\\left\(x\_\{p\}^\{\(i\)\},x\_\{q\}^\{\(i\)\}\\right\)\\right\|\\leq\\widetilde\{C\}\_\{x\}\\gamma\_\{p\-q\}\.Hence, for everyss,

𝔼​\|bs\(i\)​\(αi\)\|2≤C~x​∑j,k≥1\|Δ​ψj\|​\|Δ​ψk\|​γj−k≤C~x​‖γ‖ℓ1​\(ℤ\)​∑j≥1\|Δ​ψj\|2,\\displaystyle\\mathbb\{E\}\|b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\\leq\\widetilde\{C\}\_\{x\}\\sum\_\{j,k\\geq 1\}\|\\Delta\\psi\_\{j\}\|\|\\Delta\\psi\_\{k\}\|\\gamma\_\{j\-k\}\\leq\\widetilde\{C\}\_\{x\}\\\|\\gamma\\\|\_\{\\ell\_\{1\}\(\\mathbb\{Z\}\)\}\\sum\_\{j\\geq 1\}\|\\Delta\\psi\_\{j\}\|^\{2\},where the second inequality follows from Young’s convolution inequality\. Since

‖γ‖ℓ1​\(ℤ\)≤1\+2αmin,\\\|\\gamma\\\|\_\{\\ell\_\{1\}\(\\mathbb\{Z\}\)\}\\leq 1\+\\frac\{2\}\{\\alpha\_\{\\min\}\},the definition ofLxL\_\{x\}and \([52](https://arxiv.org/html/2609.18127#A2.E52)\) imply

qi,αi≤Lx​S12​\|hi​\(αi\)\|2\.q\_\{i,\\alpha\_\{i\}\}\\leq L\_\{x\}S\_\{1\}^\{2\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{2\}\.\(53\)
Letmi,αi:=Σx,t−1​ci,αi\.m\_\{i,\\alpha\_\{i\}\}:=\\Sigma\_\{x,t\}^\{\-1\}c\_\{i,\\alpha\_\{i\}\}\.By \([48](https://arxiv.org/html/2609.18127#A2.E48)\),

qi,αi=\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]\+mi,αi⊤​Σx,t​mi,αi\.q\_\{i,\\alpha\_\{i\}\}=\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\+m\_\{i,\\alpha\_\{i\}\}^\{\\top\}\\Sigma\_\{x,t\}m\_\{i,\\alpha\_\{i\}\}\.Thus

mi,αi⊤​Σx,t​mi,αi≤qi,αi\.m\_\{i,\\alpha\_\{i\}\}^\{\\top\}\\Sigma\_\{x,t\}m\_\{i,\\alpha\_\{i\}\}\\leq q\_\{i,\\alpha\_\{i\}\}\.Combining this with \([47](https://arxiv.org/html/2609.18127#A2.E47)\) and \([53](https://arxiv.org/html/2609.18127#A2.E53)\) yields

‖mi,αi‖2≤2​Lxσ​S1​\|hi​\(αi\)\|\.\\\|m\_\{i,\\alpha\_\{i\}\}\\\|\_\{2\}\\leq\\frac\{\\sqrt\{2L\_\{x\}\}\}\{\\sigma\}S\_\{1\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|\.\(54\)
Fix a unit vectoru∈ℝn\+1u\\in\\mathbb\{R\}^\{n\+1\}and writeΣi,αi−1/2u=\[βb\]\.\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}u=\\begin\{bmatrix\}\\beta\\\\ b\\end\{bmatrix\}\.The block factorization

Σi,αi=\[Imi,αi01\]⊤​\[Σx,t00Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]​\[Imi,αi01\]\\Sigma\_\{i,\\alpha\_\{i\}\}=\\begin\{bmatrix\}I&m\_\{i,\\alpha\_\{i\}\}\\\\ 0&1\\end\{bmatrix\}^\{\\\!\\top\}\\begin\{bmatrix\}\\Sigma\_\{x,t\}&0\\\\ 0&R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\end\{bmatrix\}\\begin\{bmatrix\}I&m\_\{i,\\alpha\_\{i\}\}\\\\ 0&1\\end\{bmatrix\}andu⊤​u=1u^\{\\top\}u=1give

1=\(β\+b​mi,αi\)⊤​Σx,t​\(β\+b​mi,αi\)\+\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]​b2\.1=\(\\beta\+bm\_\{i,\\alpha\_\{i\}\}\)^\{\\top\}\\Sigma\_\{x,t\}\(\\beta\+bm\_\{i,\\alpha\_\{i\}\}\)\+\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]b^\{2\}\.Hence, by \([47](https://arxiv.org/html/2609.18127#A2.E47)\) and \([50](https://arxiv.org/html/2609.18127#A2.E50)\),

‖β\+b​mi,αi‖2≤2σ,\|b\|≤1γ\.\\\|\\beta\+bm\_\{i,\\alpha\_\{i\}\}\\\|\_\{2\}\\leq\\frac\{\\sqrt\{2\}\}\{\\sigma\},\\qquad\|b\|\\leq\\frac\{1\}\{\\sqrt\{\\gamma\}\}\.\(55\)
Define the scalar whitened process

rs:=u⊤Σi,αi−1/2zs\(i,αi\)=β⊤xs\+bbs\(i\)\(αi\)\.r\_\{s\}:=u^\{\\top\}\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}z\_\{s\}^\{\(i,\\alpha\_\{i\}\)\}=\\beta^\{\\top\}x\_\{s\}\+b\\,b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\.It is a linear filter of the state process\. By \([51](https://arxiv.org/html/2609.18127#A2.E51)\), \([54](https://arxiv.org/html/2609.18127#A2.E54)\), and \([55](https://arxiv.org/html/2609.18127#A2.E55)\), the sum of the Euclidean norms of its filter coefficients is at most

‖β‖2\+\|b\|​∑j≥1\|Δ​ψj\|\\displaystyle\\\|\\beta\\\|\_\{2\}\+\|b\|\\sum\_\{j\\geq 1\}\|\\Delta\\psi\_\{j\}\|≤‖β\+b​mi,αi‖2\+\|b\|\(‖mi,αi‖2\+∑j≥1\|Δ​ψj\|\)\\displaystyle\\leq\\\|\\beta\+bm\_\{i,\\alpha\_\{i\}\}\\\|\_\{2\}\+\|b\|\\left\(\\\|m\_\{i,\\alpha\_\{i\}\}\\\|\_\{2\}\+\\sum\_\{j\\geq 1\}\|\\Delta\\psi\_\{j\}\|\\right\)≤2σ\+S1​\|hi​\(αi\)\|γ​\(1\+2​Lxσ\)\.\\displaystyle\\leq\\frac\{\\sqrt\{2\}\}\{\\sigma\}\+\\frac\{S\_\{1\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|\}\{\\sqrt\{\\gamma\}\}\\left\(1\+\\frac\{\\sqrt\{2L\_\{x\}\}\}\{\\sigma\}\\right\)\.Sinceαi,αi,⋆∈\(0,1\]\\alpha\_\{i\},\\alpha\_\{i,\\star\}\\in\(0,1\],\|hi​\(αi\)\|≤1\|h\_\{i\}\(\\alpha\_\{i\}\)\|\\leq 1, and therefore this coefficient norm is at mostFGF\_\{\\mathrm\{G\}\}\.

Writing the filter representation as

rs=∑ℓ≥0fℓ⊤​xs−ℓ,∑ℓ≥0‖fℓ‖2≤FG,r\_\{s\}=\\sum\_\{\\ell\\geq 0\}f\_\{\\ell\}^\{\\top\}x\_\{s\-\\ell\},\\qquad\\sum\_\{\\ell\\geq 0\}\\\|f\_\{\\ell\}\\\|\_\{2\}\\leq F\_\{\\mathrm\{G\}\},we obtain

∑k∈ℤ\|Cov⁡\(rp,rp\+k\)\|≤C~x​∑ℓ,m≥0‖fℓ‖2​‖fm‖2​∑k∈ℤγk\+ℓ−m≤Lx​FG2\.\\displaystyle\\sum\_\{k\\in\\mathbb\{Z\}\}\|\\operatorname\{Cov\}\(r\_\{p\},r\_\{p\+k\}\)\|\\leq\\widetilde\{C\}\_\{x\}\\sum\_\{\\ell,m\\geq 0\}\\\|f\_\{\\ell\}\\\|\_\{2\}\\\|f\_\{m\}\\\|\_\{2\}\\sum\_\{k\\in\\mathbb\{Z\}\}\\gamma\_\{k\+\\ell\-m\}\\leq L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}\.\(56\)LetTi,αi,uT\_\{i,\\alpha\_\{i\},u\}denote the covariance matrix of\(r0,…,rt−1\)⊤\(r\_\{0\},\\ldots,r\_\{t\-1\}\)^\{\\top\}\. The preceding covariance\-sum bound \([56](https://arxiv.org/html/2609.18127#A2.E56)\) yields

‖Ti,αi,u‖op≤Lx​FG2\.\\\|T\_\{i,\\alpha\_\{i\},u\}\\\|\_\{\\mathrm\{op\}\}\\leq L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}\.\(57\)Moreover,

tr\(Ti,αi,u\)=∑s=0t−1𝔼\[rs2\]=tu⊤Σi,αi−1/2Σi,αiΣi,αi−1/2u=t\.\\displaystyle\\operatorname\{tr\}\(T\_\{i,\\alpha\_\{i\},u\}\)=\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\[r\_\{s\}^\{2\}\]=t\\,u^\{\\top\}\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}\\Sigma\_\{i,\\alpha\_\{i\}\}\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}u=t\.SinceTi,αi,u⪰0T\_\{i,\\alpha\_\{i\},u\}\\succeq 0,

‖Ti,αi,u‖F2≤‖Ti,αi,u‖op​tr⁡\(Ti,αi,u\)≤t​Lx​FG2\.\\\|T\_\{i,\\alpha\_\{i\},u\}\\\|\_\{F\}^\{2\}\\leq\\\|T\_\{i,\\alpha\_\{i\},u\}\\\|\_\{\\mathrm\{op\}\}\\operatorname\{tr\}\(T\_\{i,\\alpha\_\{i\},u\}\)\\leq tL\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}\.\(58\)
The vector\(r0,…,rt−1\)⊤\(r\_\{0\},\\ldots,r\_\{t\-1\}\)^\{\\top\}is jointly Gaussian\. Thus, for everyx\>0x\>0, the Gaussian quadratic\-form inequality with \([57](https://arxiv.org/html/2609.18127#A2.E57)\) and \([58](https://arxiv.org/html/2609.18127#A2.E58)\) gives

\|1t​∑s=0t−1rs2−1\|≤2​Lx​FG2​xt\+2​Lx​FG2​xt\\left\|\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}r\_\{s\}^\{2\}\-1\\right\|\\leq 2\\sqrt\{\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}x\}\{t\}\}\+2\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}x\}\{t\}with probability at least1−2​e−x1\-2e^\{\-x\}\. Equivalently,

\|u⊤Σi,αi−1/2\(Σ^i,αi−Σi,αi\)Σi,αi−1/2u\|≤2Lx​FG2​xt\+2Lx​FG2​xt\.\\left\|u^\{\\top\}\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}\\left\(\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}\-\\Sigma\_\{i,\\alpha\_\{i\}\}\\right\)\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}u\\right\|\\leq 2\\sqrt\{\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}x\}\{t\}\}\+2\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}x\}\{t\}\.
Apply this inequality to a1/41/4\-net of the unit sphere inℝn\+1\\mathbb\{R\}^\{n\+1\}, whose cardinality is at most9n\+19^\{n\+1\}, and take a union bound over all grid points\. Their total number is at most∑i=1nMi\\sum\_\{i=1\}^\{n\}M\_\{i\}\. Takingx=qG​\(δ\)x=q\_\{\\mathrm\{G\}\}\(\\delta\)gives, with probability at least1−δ1\-\\delta, simultaneously for alliiandαi∈𝒜ϵ,i\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}satisfying\|hi​\(αi\)\|\>ri,sep\|h\_\{i\}\(\\alpha\_\{i\}\)\|\>r\_\{\\mathrm\{i,sep\}\},

‖Σi,αi−1/2\(Σ^i,αi−Σi,αi\)Σi,αi−1/2‖op≤4\(Lx​FG2​qG​\(δ\)t\+Lx​FG2​qG​\(δ\)t\)\.\\left\\\|\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}\\left\(\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}\-\\Sigma\_\{i,\\alpha\_\{i\}\}\\right\)\\Sigma\_\{i,\\alpha\_\{i\}\}^\{\-1/2\}\\right\\\|\_\{\\mathrm\{op\}\}\\leq 4\\left\(\\sqrt\{\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}q\_\{\\mathrm\{G\}\}\(\\delta\)\}\{t\}\}\+\\frac\{L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}q\_\{\\mathrm\{G\}\}\(\\delta\)\}\{t\}\\right\)\.\(59\)If

t≥256​Lx​FG2​qG​\(δ\),t\\geq 256L\_\{x\}F\_\{\\mathrm\{G\}\}^\{2\}q\_\{\\mathrm\{G\}\}\(\\delta\),then the right\-hand side of \([59](https://arxiv.org/html/2609.18127#A2.E59)\) is at most

4​\(116\+1256\)=1764<12\.4\\left\(\\frac\{1\}\{16\}\+\\frac\{1\}\{256\}\\right\)=\\frac\{17\}\{64\}<\\frac\{1\}\{2\}\.Therefore

Σ^i,αi⪰12​Σi,αi\.\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}\\succeq\\frac\{1\}\{2\}\\Sigma\_\{i,\\alpha\_\{i\}\}\.\(60\)
Finally, \([60](https://arxiv.org/html/2609.18127#A2.E60)\) implies, for everyvv,

w​\(v\)⊤​Σ^i,αi​w​\(v\)≥12​w​\(v\)⊤​Σi,αi​w​\(v\)\.w\(v\)^\{\\top\}\\hat\{\\Sigma\}\_\{i,\\alpha\_\{i\}\}w\(v\)\\geq\\frac\{1\}\{2\}w\(v\)^\{\\top\}\\Sigma\_\{i,\\alpha\_\{i\}\}w\(v\)\.Taking the infimum overvvand using \([46](https://arxiv.org/html/2609.18127#A2.E46)\) and \([49](https://arxiv.org/html/2609.18127#A2.E49)\),

1t​𝒬¯t,i​\(αi\)≥12​\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]\.\\frac\{1\}\{t\}\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{1\}\{2\}\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\.Thus

𝒬¯t,i​\(αi\)≥t2​\[Rt\(i\)​\(αi\)−Rt\(i\)​\(αi,⋆\)\]\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\}\{2\}\\left\[R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\.Finally, \([50](https://arxiv.org/html/2609.18127#A2.E50)\) gives

𝒬¯t,i​\(αi\)≥t​γ2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\alpha\_\{i\}\)\\geq\\frac\{t\\gamma\}\{2\}\.∎

###### Lemma 13\(Concentration of the Taylor remainder\)\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds\. Let

ℒρ,i:=14​\(9​e4​ℨi\)2​ℒx\.\\mathcal\{L\}\_\{\\rho,i\}:=\\frac\{1\}\{4\}\\left\(\\frac\{9e\}\{4\}\\,\\mathfrak\{Z\}\_\{i\}\\right\)^\{2\}\\mathcal\{L\}\_\{x\}\.Define

Mloc:=∑i=1n\|𝒜ϵ,i∩\[α¯i,loc,1\]\|,q2​\(δ\):=log⁡2​Mlocδ\.M\_\{\\mathrm\{loc\}\}:=\\sum\_\{i=1\}^\{n\}\\left\|\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\]\\right\|,\\qquad q\_\{2\}\(\\delta\):=\\log\\frac\{2M\_\{\\mathrm\{loc\}\}\}\{\\delta\}\.Then, with probability at least1−δ/21\-\\delta/2, simultaneously for alli∈\[n\]i\\in\[n\]and allαi∈𝒜ϵ,i∩\[α¯i,loc,1\]\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\],

1t​∑s=0t−1\|ρs\(i\)​\(αi\)\|2≤\|hi​\(αi\)\|4​\[Ki24\+2​ℒρ,i​\(q2​\(δ\)t\+q2​\(δ\)t\)\]\.\\displaystyle\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\\leq\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\}\\left\[\\frac\{K\_\{i\}^\{2\}\}\{4\}\+2\\mathcal\{L\}\_\{\\rho,i\}\\left\(\\sqrt\{\\frac\{q\_\{2\}\(\\delta\)\}\{t\}\}\+\\frac\{q\_\{2\}\(\\delta\)\}\{t\}\\right\)\\right\]\.\(61\)Consequently, if

t≥tE2​\(δ\):=q2​\(δ\)​max⁡\{16​ℒx2σ4​C~G4​ζ​\(2\)2,4​ℒxσ2​C~G2​ζ​\(2\)\},t\\geq t\_\{\\mathrm\{E2\}\}\(\\delta\):=q\_\{2\}\(\\delta\)\\max\\left\\\{\\frac\{16\\mathcal\{L\}\_\{x\}^\{2\}\}\{\\sigma^\{4\}\\widetilde\{C\}\_\{G\}^\{4\}\\zeta\(2\)^\{2\}\},\\frac\{4\\mathcal\{L\}\_\{x\}\}\{\\sigma^\{2\}\\widetilde\{C\}\_\{G\}^\{2\}\\zeta\(2\)\}\\right\\\},\(62\)then event\(E2\)holds simultaneously over all local grid points with probability at least1−δ/21\-\\delta/2\.

###### Proof\.

Fixiiandαi∈𝒜ϵ,i∩\[α¯i,loc,1\]\\alpha\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\underline\{\\alpha\}\_\{i,\\mathrm\{loc\}\},1\]\. For

qi,j​\(αi\):=ψ⁡\(αi,j\)−ψ⁡\(αi,⋆,j\)−hi​\(αi\)​∂αψ⁡\(αi,⋆,j\),q\_\{i,j\}\(\\alpha\_\{i\}\):=\\psi\(\\alpha\_\{i\},j\)\-\\psi\(\\alpha\_\{i,\\star\},j\)\-h\_\{i\}\(\\alpha\_\{i\}\)\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\),the integral Taylor formula gives

qi,j​\(αi\)=∫αi,⋆αi\(αi−u\)​∂u2ψ⁡\(u,j\)​𝑑u,ρs\(i\)​\(αi\)=∑j≥1qi,j​\(αi\)​xs\+1−j\(i\)\.q\_\{i,j\}\(\\alpha\_\{i\}\)=\\int\_\{\\alpha\_\{i,\\star\}\}^\{\\alpha\_\{i\}\}\(\\alpha\_\{i\}\-u\)\\partial\_\{u\}^\{2\}\\psi\(u,j\)\\,\\mathrm\{d\}u,\\qquad\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)=\\sum\_\{j\\geq 1\}q\_\{i,j\}\(\\alpha\_\{i\}\)x\_\{s\+1\-j\}^\{\(i\)\}\.Therefore, by \([33](https://arxiv.org/html/2609.18127#A2.E33)\),

∑j≥1\|qi,j​\(αi\)\|≤9​e8​ℨi​\|hi​\(αi\)\|2\.\\sum\_\{j\\geq 1\}\|q\_\{i,j\}\(\\alpha\_\{i\}\)\|\\leq\\frac\{9e\}\{8\}\\,\\mathfrak\{Z\}\_\{i\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{2\}\.\(63\)Using \([63](https://arxiv.org/html/2609.18127#A2.E63)\) and the same convolution calculation as in Lemma[11](https://arxiv.org/html/2609.18127#Thmlemma11),

supp∑k∈ℤ\|Cov⁡\(ρp\(i\)​\(αi\),ρp\+k\(i\)​\(αi\)\)\|≤ℒρ,i​\|hi​\(αi\)\|4\.\\sup\_\{p\}\\sum\_\{k\\in\\mathbb\{Z\}\}\|\\operatorname\{Cov\}\(\\rho\_\{p\}^\{\(i\)\}\(\\alpha\_\{i\}\),\\rho\_\{p\+k\}^\{\(i\)\}\(\\alpha\_\{i\}\)\)\|\\leq\\mathcal\{L\}\_\{\\rho,i\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\}\.Hence, forρi​\(αi\)=\(ρ0\(i\)​\(αi\),…,ρt−1\(i\)​\(αi\)\)⊤∼𝒩⁡\(0,ℛi,αi\)\\rho\_\{i\}\(\\alpha\_\{i\}\)=\(\\rho\_\{0\}^\{\(i\)\}\(\\alpha\_\{i\}\),\\ldots,\\rho\_\{t\-1\}^\{\(i\)\}\(\\alpha\_\{i\}\)\)^\{\\top\}\\sim\\mathcal\{N\}\(0,\\mathcal\{R\}\_\{i,\\alpha\_\{i\}\}\),

‖ℛi,αi‖op≤ℒρ,i​\|hi​\(αi\)\|4,‖ℛi,αi‖F≤t​ℒρ,i​\|hi​\(αi\)\|4\.\\\|\\mathcal\{R\}\_\{i,\\alpha\_\{i\}\}\\\|\_\{\\mathrm\{op\}\}\\leq\\mathcal\{L\}\_\{\\rho,i\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\},\\qquad\\\|\\mathcal\{R\}\_\{i,\\alpha\_\{i\}\}\\\|\_\{F\}\\leq\\sqrt\{t\}\\,\\mathcal\{L\}\_\{\\rho,i\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\}\.The one\-sided Gaussian quadratic\-form inequality gives, with probability at least1−e−x1\-e^\{\-x\},

1t​∑s=0t−1\|ρs\(i\)​\(αi\)\|2\\displaystyle\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}≤1t​∑s=0t−1𝔼​\|ρs\(i\)​\(αi\)\|2\\displaystyle\\leq\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\|\\rho\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)\|^\{2\}\+2​ℒρ,i​\|hi​\(αi\)\|4​\(xt\+xt\)\.\\displaystyle\\quad\+2\\mathcal\{L\}\_\{\\rho,i\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\}\\left\(\\sqrt\{\\frac\{x\}\{t\}\}\+\\frac\{x\}\{t\}\\right\)\.Lemma[9](https://arxiv.org/html/2609.18127#Thmlemma9)bounds the expectation term byKi2​\|hi​\(αi\)\|4/4K\_\{i\}^\{2\}\|h\_\{i\}\(\\alpha\_\{i\}\)\|^\{4\}/4\. Takingx=q2​\(δ\)x=q\_\{2\}\(\\delta\)and applying a union bound proves \([61](https://arxiv.org/html/2609.18127#A2.E61)\)\. Finally, \([62](https://arxiv.org/html/2609.18127#A2.E62)\) implies

2​ℒρ,i​\(q2​\(δ\)t\+q2​\(δ\)t\)≤Ki24,2\\mathcal\{L\}\_\{\\rho,i\}\\left\(\\sqrt\{\\frac\{q\_\{2\}\(\\delta\)\}\{t\}\}\+\\frac\{q\_\{2\}\(\\delta\)\}\{t\}\\right\)\\leq\\frac\{K\_\{i\}^\{2\}\}\{4\},which proves event \(E2\)\. ∎

We now have all the tools needed to prove[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)\. We begin by restating[Theorem1](https://arxiv.org/html/2609.18127#Thmtheorem1)in its full finite sample form for the estimation error of𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}\.

###### Theorem 3\(Full error bound for𝜶⋆\\boldsymbol\{\\alpha\}\_\{\\star\}\)\.

Under[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1), defineϵmax:=max1≤i≤n⁡ϵi,\\epsilon\_\{\\max\}:=\\max\_\{1\\leq i\\leq n\}\\epsilon\_\{i\},andμmin:=min1≤i≤n⁡μt,i\.\\mu\_\{\\min\}:=\\min\_\{1\\leq i\\leq n\}\\mu\_\{t,i\}\.Let

γ:=min1≤i≤n​infα∈𝒜ϵ,i,\|αi−αi,⋆\|\>μi,lb2​Ki\{Rt\(i\)​\(αi\)−Rt\(i\)​\(αi∘\)\}\\gamma:=\\underset\{1\\leq i\\leq n\}\{\\min\}\\inf\_\{\\alpha\\in\\mathcal\{A\}\_\{\\epsilon,i\},\\,\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\>\\frac\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\{2K\_\{i\}\}\}\\left\\\{R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}^\{\\circ\}\)\\right\\\}\(64\)and supposeγ\>0\\gamma\>0\. Fixδ∈\(0,1/2\)\\delta\\in\(0,1/2\)and choose0<θ≤min⁡\{12,δ​μmin576​σ2​n​S12​tr⁡\(Γt\)\}\.0<\\theta\\leq\\min\\left\\\{\\frac\{1\}\{2\},\\,\\frac\{\\delta\\mu\_\{\\min\}\}\{576\\sigma^\{2\}nS\_\{1\}^\{2\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\\right\\\}\.Define

𝔅⁡\(θ,δ,k\):=\\displaystyle\\mathfrak\{B\}\(\\theta,\\delta,k\):=18​θ​σ2​n​S12​t​tr⁡\(Γt\)δ\+27​σ2​n​S12​t​tr⁡\(Γt\)δ​ϵmax2\\displaystyle\\frac\{18\\theta\\sigma^\{2\}nS\_\{1\}^\{2\}t\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\+\\frac\{27\\sigma^\{2\}nS\_\{1\}^\{2\}t\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\\epsilon\_\{\\max\}^\{2\}\+8​σ2θ∑i=1nlog9​Mi​nδ\+8σ2log9δ\+36​\(1\+θ−1\)​C2​σ2​n​tr⁡\(Γt\)δ​λmin​\(Γk\)Ξt\(δ,k\),\\displaystyle\+\\frac\{8\\sigma^\{2\}\}\{\\theta\}\\sum\_\{i=1\}^\{n\}\\log\\frac\{9M\_\{i\}n\}\{\\delta\}\+8\\sigma^\{2\}\\log\\frac\{9\}\{\\delta\}\+\\frac\{36\(1\+\\theta^\{\-1\}\)C^\{2\}\\sigma^\{2\}n\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\),whereΞt\(δ,k\):=nlog9​nδ\+logdet\(ΓtΓk−1\)\.\\Xi\_\{t\}\(\\delta,k\):=n\\log\\frac\{9n\}\{\\delta\}\+\\log\\det\\\!\\left\(\\Gamma\_\{t\}\\Gamma\_\{k\}^\{\-1\}\\right\)\.Ift≳1αmin4​\(n\+log⁡∑i=1nMiδ\),t\\gtrsim\\frac\{1\}\{\\alpha\_\{\\min\}^\{4\}\}\\left\(n\+\\log\\frac\{\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\\right\),and that, for some integerkk,

tk≥c​Ξt​\(δ,k\)\.\\frac\{t\}\{k\}\\geq c\\,\\Xi\_\{t\}\(\\delta,k\)\.\(65\)Suppose that

𝔅⁡\(θ,δ,k\)<t​γ2\.\\mathfrak\{B\}\(\\theta,\\delta,k\)<\\frac\{t\\gamma\}\{2\}\.\(66\)Then, with probability at least1−δ1\-\\delta,

‖𝜶^−𝜶⋆‖∞2≤\\displaystyle\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\leq\{\}288​σ2​n​S12​tr⁡\(Γt\)δ​μmin​ϵmax2\\displaystyle\\frac\{288\\sigma^\{2\}nS\_\{1\}^\{2\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\mu\_\{\\min\}\}\\epsilon\_\{\\max\}^\{2\}\+2563​t​μmin​θ​\[σ2​\(∑i=1nlog⁡9​Mi​nδ\+log⁡9δ\)\+9​C2​σ2​n​tr⁡\(Γt\)δ​λmin​\(Γk\)​Ξt​\(δ,k\)\]\.\\displaystyle\+\\frac\{256\}\{3t\\mu\_\{\\min\}\\theta\}\\Bigg\[\\sigma^\{2\}\\left\(\\sum\_\{i=1\}^\{n\}\\log\\frac\{9M\_\{i\}n\}\{\\delta\}\+\\log\\frac\{9\}\{\\delta\}\\right\)\+\\frac\{9C^\{2\}\\sigma^\{2\}n\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\)\\Bigg\]\.\(67\)Consequently, ifϵmax=𝒪\(t−1/2\),\\epsilon\_\{\\max\}=\\mathcal\{O\}\\left\(t^\{\-1/2\}\\right\),then‖𝛂^−𝛂⋆‖∞=𝒪\(t−1/2\)\.\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\right\\\|\_\{\\infty\}=\\mathcal\{O\}\\left\(t^\{\-1/2\}\\right\)\.

###### Proof\.

We intersect three events: the global profiled lower isometry event from[Lemma12](https://arxiv.org/html/2609.18127#Thmlemma12), the local lower isometry event from[Lemma10](https://arxiv.org/html/2609.18127#Thmlemma10), and the in\-sample event from[Lemma8](https://arxiv.org/html/2609.18127#Thmlemma8), with failure probabilityδ/3\\delta/3allocated to each\. Byt≳1αmin4​\(n\+log⁡∑i=1nMiδ\)t\\gtrsim\\frac\{1\}\{\\alpha\_\{\\min\}^\{4\}\}\\left\(n\+\\log\\frac\{\\sum\_\{i=1\}^\{n\}M\_\{i\}\}\{\\delta\}\\right\), their intersection has probability at least1−δ1\-\\delta\. We work on this intersection throughout\.

Step 1: Global localization\.Suppose, toward a contradiction, that for somei∈\[n\]i\\in\[n\],

\|α^i−αi,⋆\|\>ri,sep\.\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|\>r\_\{i,\\mathrm\{sep\}\}\.Since the global profiled lower\-isometry event holds simultaneously over all grid points outside the separation neighborhood,

𝒬¯t,i​\(α^i\)≥t2​\[Rt\(i\)​\(α^i\)−Rt\(i\)​\(αi,⋆\)\]≥t​γ2\.\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\\geq\\frac\{t\}\{2\}\\left\[R\_\{t\}^\{\(i\)\}\(\\hat\{\\alpha\}\_\{i\}\)\-R\_\{t\}^\{\(i\)\}\(\\alpha\_\{i,\\star\}\)\\right\]\\geq\\frac\{t\\gamma\}\{2\}\.Recall

ℰt=∑s=0t−1‖\(Δ𝜶^−Δ𝜶⋆\)​xs\+1−\(A^−A⋆\)​xs‖22=∑j=1n𝒬t,j​\(α^j,a^j−aj,⋆\)\.\\displaystyle\\mathcal\{E\}\_\{t\}=\\sum\_\{s=0\}^\{t\-1\}\\left\\\|\\left\(\\Delta^\{\\hat\{\\boldsymbol\{\\alpha\}\}\}\-\\Delta^\{\\boldsymbol\{\\alpha\}^\{\\star\}\}\\right\)x\_\{s\+1\}\-\(\\hat\{A\}\-A\_\{\\star\}\)x\_\{s\}\\right\\\|\_\{2\}^\{2\}=\\sum\_\{j=1\}^\{n\}\\mathcal\{Q\}\_\{t,j\}\\bigl\(\\hat\{\\alpha\}\_\{j\},\\hat\{a\}\_\{j\}\-a\_\{j,\\star\}\\bigr\)\.Since every term in the sum is nonnegative and𝒬¯t,i\\underline\{\\mathcal\{Q\}\}\_\{t,i\}is obtained by profiling over the row parameter,

ℰt≥𝒬t,i​\(α^i,a^i−ai,⋆\)≥𝒬¯t,i​\(α^i\)≥t​γ2\.\\mathcal\{E\}\_\{t\}\\geq\\mathcal\{Q\}\_\{t,i\}\\bigl\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\bigr\)\\geq\\underline\{\\mathcal\{Q\}\}\_\{t,i\}\(\\hat\{\\alpha\}\_\{i\}\)\\geq\\frac\{t\\gamma\}\{2\}\.\(68\)
We next obtain the corresponding upper bound from[Lemma8](https://arxiv.org/html/2609.18127#Thmlemma8)\. Setτ=ρ=θ\\tau=\\rho=\\thetaand use failure probabilityδ/3\\delta/3\. The Lemma 3 good event retained in its proof also gives

Xt​Xt⊤⪯9​σ2​n​tδ​Γt,tr⁡\(Xt​Xt⊤\)≤9​σ2​n​tδ​tr⁡\(Γt\)\.X\_\{t\}X\_\{t\}^\{\\top\}\\preceq\\frac\{9\\sigma^\{2\}nt\}\{\\delta\}\\Gamma\_\{t\},\\qquad\\operatorname\{tr\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\\leq\\frac\{9\\sigma^\{2\}nt\}\{\\delta\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\.\(69\)Moreover, since both the true and estimated fractional orders lie in\(0,1\]\(0,1\],

‖𝜶^−𝜶⋆‖∞≤1\.\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}\\leq 1\.Substituting these two bounds into[Lemma8](https://arxiv.org/html/2609.18127#Thmlemma8)yields

ℰt≤𝔅⁡\(θ,δ,k\)\.\\mathcal\{E\}\_\{t\}\\leq\\mathfrak\{B\}\(\\theta,\\delta,k\)\.By𝔅⁡\(θ,δ,k\)<t​γ2\\mathfrak\{B\}\(\\theta,\\delta,k\)<\\frac\{t\\gamma\}\{2\},

ℰt<t​γ2,\\mathcal\{E\}\_\{t\}<\\frac\{t\\gamma\}\{2\},contradicting \([68](https://arxiv.org/html/2609.18127#A2.E68)\)\. Therefore,

\|α^i−αi,⋆\|≤ri,sep,i∈\[n\]\.\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|\\leq r\_\{i,\\mathrm\{sep\}\},\\qquad i\\in\[n\]\.\(70\)
We now verify that this coarse localization places the estimator in the local sets𝒢i\\mathcal\{G\}\_\{i\}\. Recall that

ℨi=∑j≥1\(1\+log⁡j\)2j1\+αi,⋆/2≥ζ⁡\(1\+αi,⋆2\)≥2αi,⋆\.\\mathfrak\{Z\}\_\{i\}=\\sum\_\{j\\geq 1\}\\frac\{\(1\+\\log j\)^\{2\}\}\{j^\{1\+\\alpha\_\{i,\\star\}/2\}\}\\geq\\zeta\\\!\\left\(1\+\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\\right\)\\geq\\frac\{2\}\{\\alpha\_\{i,\\star\}\}\.SinceG0=InG\_\{0\}=I\_\{n\},C~G≥1\\widetilde\{C\}\_\{G\}\\geq 1\. Moreover,\|di,2\|≤1/2\|d\_\{i,2\}\|\\leq 1/2and\|di,3\|≤1/3\|d\_\{i,3\}\|\\leq 1/3, soκi,⋆≤13/36\\kappa\_\{i,\\star\}\\leq 13/36\. Hence

ri,sep≤αi,⋆​13108​e​ζ⁡\(2\)<αi,⋆2\.r\_\{i,\\mathrm\{sep\}\}\\leq\\frac\{\\alpha\_\{i,\\star\}\\sqrt\{13\}\}\{108e\\sqrt\{\\zeta\(2\)\}\}<\\frac\{\\alpha\_\{i,\\star\}\}\{2\}\.Together with \([70](https://arxiv.org/html/2609.18127#A2.E70)\), this implies

α^i\>αi,⋆2=αi,loc\.\\hat\{\\alpha\}\_\{i\}\>\\frac\{\\alpha\_\{i,\\star\}\}\{2\}=\\alpha\_\{i,\\mathrm\{loc\}\}\.Sinceα^i∈𝒜ϵ,i⊂\(0,1\]\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\subset\(0,1\],

α^i∈𝒜ϵ,i∩\[αi,loc,1\]\.\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{A\}\_\{\\epsilon,i\}\\cap\[\\alpha\_\{i,\\mathrm\{loc\}\},1\]\.Furthermore, the population curvature lower bound gives

μt,i≥μi,lb,ri,sep=μi,lb2​Ki\.\\mu\_\{t,i\}\\geq\\mu\_\{i,\\mathrm\{lb\}\},\\qquad r\_\{i,\\mathrm\{sep\}\}=\\frac\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\{2K\_\{i\}\}\.Hence

\|α^i−αi,⋆\|≤μi,lb2​Ki,\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|\\leq\\frac\{\\sqrt\{\\mu\_\{i,\\mathrm\{lb\}\}\}\}\{2K\_\{i\}\},and therefore

α^i∈𝒢i,i∈\[n\]\.\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\},\\qquad i\\in\[n\]\.Step 2: Local lower isometry\.Sinceα^i∈𝒢i\\hat\{\\alpha\}\_\{i\}\\in\\mathcal\{G\}\_\{i\}for every row,[Lemma10](https://arxiv.org/html/2609.18127#Thmlemma10)gives

ℰt\\displaystyle\\mathcal\{E\}\_\{t\}=∑i=1n𝒬t,i​\(α^i,a^i−ai,⋆\)\\displaystyle=\\sum\_\{i=1\}^\{n\}\\mathcal\{Q\}\_\{t,i\}\\bigl\(\\hat\{\\alpha\}\_\{i\},\\hat\{a\}\_\{i\}\-a\_\{i,\\star\}\\bigr\)≥t8​∑i=1nμt,i​\|α^i−αi,⋆\|2\\displaystyle\\geq\\frac\{t\}\{8\}\\sum\_\{i=1\}^\{n\}\\mu\_\{t,i\}\|\\hat\{\\alpha\}\_\{i\}\-\\alpha\_\{i,\\star\}\|^\{2\}≥t​μmin8​‖𝜶^−𝜶⋆‖∞2\.\\displaystyle\\geq\\frac\{t\\mu\_\{\\min\}\}\{8\}\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\.\(71\)Step 3: In\-sample upper bound and absorption\.We reuse the same in\-sample event from Step 1; no additional failure probability is required\. Settingτ=ρ=θ\\tau=\\rho=\\thetain[Lemma8](https://arxiv.org/html/2609.18127#Thmlemma8)and substituting \([69](https://arxiv.org/html/2609.18127#A2.E69)\) gives

ℰt≤\\displaystyle\\mathcal\{E\}\_\{t\}\\leq\{\}18​θ​σ2​n​S12​t​tr⁡\(Γt\)δ​‖𝜶^−𝜶⋆‖∞2\\displaystyle\\frac\{18\\theta\\sigma^\{2\}nS\_\{1\}^\{2\}t\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\+27​σ2​n​S12​t​tr⁡\(Γt\)δ​ϵmax2\\displaystyle\+\\frac\{27\\sigma^\{2\}nS\_\{1\}^\{2\}t\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\\epsilon\_\{\\max\}^\{2\}\+8​σ2θ∑i=1nlog9​Mi​nδ\+8σ2log9δ\\displaystyle\+\\frac\{8\\sigma^\{2\}\}\{\\theta\}\\sum\_\{i=1\}^\{n\}\\log\\frac\{9M\_\{i\}n\}\{\\delta\}\+8\\sigma^\{2\}\\log\\frac\{9\}\{\\delta\}\+36​\(1\+θ−1\)​C2​σ2​n​tr⁡\(Γt\)δ​λmin​\(Γk\)​Ξt​\(δ,k\)\.\\displaystyle\+\\frac\{36\(1\+\\theta^\{\-1\}\)C^\{2\}\\sigma^\{2\}n\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\)\.\(72\)
By0<θ≤min⁡\{12,δ​μmin576​σ2​n​S12​tr⁡\(Γt\)\},0<\\theta\\leq\\min\\left\\\{\\frac\{1\}\{2\},\\,\\frac\{\\delta\\mu\_\{\\min\}\}\{576\\sigma^\{2\}nS\_\{1\}^\{2\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\\right\\\},

18​θ​σ2​n​S12​tr⁡\(Γt\)δ≤μmin32\.\\frac\{18\\theta\\sigma^\{2\}nS\_\{1\}^\{2\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\\leq\\frac\{\\mu\_\{\\min\}\}\{32\}\.Combining \([71](https://arxiv.org/html/2609.18127#A2.E71)\) and \([72](https://arxiv.org/html/2609.18127#A2.E72)\) yields

3​t​μmin32​‖𝜶^−𝜶⋆‖∞2≤\\displaystyle\\frac\{3t\\mu\_\{\\min\}\}\{32\}\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\leq\{\}27​σ2​n​S12​t​tr⁡\(Γt\)δ​ϵmax2\\displaystyle\\frac\{27\\sigma^\{2\}nS\_\{1\}^\{2\}t\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\}\\epsilon\_\{\\max\}^\{2\}\+8​σ2θ∑i=1nlog9​Mi​nδ\+8σ2log9δ\\displaystyle\+\\frac\{8\\sigma^\{2\}\}\{\\theta\}\\sum\_\{i=1\}^\{n\}\\log\\frac\{9M\_\{i\}n\}\{\\delta\}\+8\\sigma^\{2\}\\log\\frac\{9\}\{\\delta\}\+36​\(1\+θ−1\)​C2​σ2​n​tr⁡\(Γt\)δ​λmin​\(Γk\)​Ξt​\(δ,k\)\.\\displaystyle\+\\frac\{36\(1\+\\theta^\{\-1\}\)C^\{2\}\\sigma^\{2\}n\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\)\.\(73\)
Sinceθ≤1/2\\theta\\leq 1/2,

1\+θ−1≤2θ,1≤1θ\.1\+\\theta^\{\-1\}\\leq\\frac\{2\}\{\\theta\},\\qquad 1\\leq\\frac\{1\}\{\\theta\}\.Dividing \([73](https://arxiv.org/html/2609.18127#A2.E73)\) by3​t​μmin/323t\\mu\_\{\\min\}/32therefore gives

‖𝜶^−𝜶⋆‖∞2≤\\displaystyle\\left\\\|\\hat\{\\boldsymbol\{\\alpha\}\}\-\\boldsymbol\{\\alpha\}^\{\\star\}\\right\\\|\_\{\\infty\}^\{2\}\\leq\{\}288​σ2​n​S12​tr⁡\(Γt\)δ​μmin​ϵmax2\\displaystyle\\frac\{288\\sigma^\{2\}nS\_\{1\}^\{2\}\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\mu\_\{\\min\}\}\\epsilon\_\{\\max\}^\{2\}\+2563​t​μmin​θ\[σ2\(∑i=1nlog9​Mi​nδ\+log9δ\)\\displaystyle\+\\frac\{256\}\{3t\\mu\_\{\\min\}\\theta\}\\Bigg\[\\sigma^\{2\}\\left\(\\sum\_\{i=1\}^\{n\}\\log\\frac\{9M\_\{i\}n\}\{\\delta\}\+\\log\\frac\{9\}\{\\delta\}\\right\)\+9​C2​σ2​n​tr⁡\(Γt\)δ​λmin​\(Γk\)Ξt\(δ,k\)\],\\displaystyle\\hskip 51\.21504pt\+\\frac\{9C^\{2\}\\sigma^\{2\}n\\operatorname\{tr\}\(\\Gamma\_\{t\}\)\}\{\\delta\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\\Xi\_\{t\}\(\\delta,k\)\\Bigg\],\(74\)which proves \([67](https://arxiv.org/html/2609.18127#A2.E67)\)\. ∎

## Appendix CProof of[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2)

###### Proof\.

[Theorem2](https://arxiv.org/html/2609.18127#Thmtheorem2)follows immediately by applying[Lemmas3](https://arxiv.org/html/2609.18127#Thmlemma3)and[4](https://arxiv.org/html/2609.18127#Thmlemma4)to \([12](https://arxiv.org/html/2609.18127#S5.E12)\)\. ∎

### C\.1Proof of[Lemma3](https://arxiv.org/html/2609.18127#Thmlemma3)

###### Proof\.

∥WtXt⊤\(XtXt⊤\)−1∥=∥Wt​Xt⊤​\(Xt​Xt⊤\)−12⏟:=T1\(Xt​Xt⊤\)−12⏟:=T2∥\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|=\\bigl\\\|\\underbrace\{W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-\\frac\{1\}\{2\}\}\}\_\{:=T\_\{1\}\}\\underbrace\{\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-\\frac\{1\}\{2\}\}\}\_\{:=T\_\{2\}\}\\bigr\\\|We first prove the processxtx\_\{t\}satisfies a block martingale small\-ball condition \(similar to Proposition 3\.1\[pmlr\-v75\-simchowitz18a\]\)\. Letℱs:=σ\(ητ:τ≤s−1\)\\mathcal\{F\}\_\{s\}:=\\sigma\(\\eta\_\{\\tau\}:\\tau\\leq s\-1\)\. For anyl≥1l\\geq 1,

xs\+l=∑m=0l−1Gm​ηs\+l−1−m\+∑m=l∞Gm​ηs\+l−1−m,x\_\{s\+l\}=\\sum\_\{m=0\}^\{l\-1\}G\_\{m\}\\eta\_\{s\+l\-1\-m\}\+\\sum\_\{m=l\}^\{\\infty\}G\_\{m\}\\eta\_\{s\+l\-1\-m\},so

xs\+l\|ℱs∼𝒩\(0,σ2Hl\),Hl=∑m=0l−1GmGm⊤\.x\_\{s\+l\}\|\\mathcal\{F\}\_\{s\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}H\_\{l\}\),\\qquad H\_\{l\}=\\sum\_\{m=0\}^\{l\-1\}G\_\{m\}G\_\{m\}^\{\\top\}\.For anyw∈𝕊n−1w\\in\\mathbb\{S\}^\{n\-1\},

⟨w,xs\+l⟩\|ℱs∼𝒩⁡\(0,σ2​wT​Hl​w\)\.\\langle w,x\_\{s\+l\}\\rangle\|\\mathcal\{F\}\_\{s\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}w^\{T\}H\_\{l\}w\)\.Since \(i\)Hℓ⪰Hk′H\_\{\\ell\}\\succeq H\_\{k^\{\\prime\}\}forℓ≥k′\\ell\\geq k^\{\\prime\}, \(ii\) Paley\-Zygmund lower bound, we have

ℙ⁡\(\|⟨w,xs\+ℓ⟩\|≥σ​w⊤​Hk′​w\|ℱs\)≥310,ℓ≥k′\.\\mathbb\{P\}\\\!\\left\(\\left\|\\langle w,x\_\{s\+\\ell\}\\rangle\\right\|\\geq\\sigma\\sqrt\{w^\{\\top\}H\_\{k^\{\\prime\}\}w\}\\,\\middle\|\\,\\mathcal\{F\}\_\{s\}\\right\)\\geq\\frac\{3\}\{10\},\\qquad\\ell\\geq k^\{\\prime\}\.Thus,

1k​∑l=1kℙ⁡\(\|⟨w,xs\+l⟩\|≥σ​w⊤​Hk′​w\)≥1k​∑l=k′kℙ⁡\(\|⟨w,xs\+l⟩\|≥σ​w⊤​Hk′​w\)≥310⋅k−k′\+1k\.\\frac\{1\}\{k\}\\sum\_\{l=1\}^\{k\}\\mathbb\{P\}\\\!\\left\(\\left\|\\langle w,x\_\{s\+l\}\\rangle\\right\|\\geq\\sigma\\sqrt\{w^\{\\top\}H\_\{k^\{\\prime\}\}w\}\\right\)\\;\\geq\\;\\frac\{1\}\{k\}\\sum\_\{l=k^\{\\prime\}\}^\{k\}\\mathbb\{P\}\\\!\\left\(\\left\|\\langle w,x\_\{s\+l\}\\rangle\\right\|\\geq\\sigma\\sqrt\{w^\{\\top\}H\_\{k^\{\\prime\}\}w\}\\right\)\\;\\geq\\;\\frac\{3\}\{10\}\\cdot\\frac\{k\-k^\{\\prime\}\+1\}\{k\}\.Pickk′=⌊k/2⌋k^\{\\prime\}=\\lfloor k/2\\rfloor\. Thereforextx\_\{t\}satisfies a\(k,Γsb,p\)\(k,\\Gamma\_\{\\mathrm\{sb\}\},p\)\-BMSB condition withΓsb=σ2​H⌊k/2⌋,p=320\.\\Gamma\_\{\\mathrm\{sb\}\}=\\sigma^\{2\}H\_\{\\lfloor k/2\\rfloor\},p=\\frac\{3\}\{20\}\. LetZi=⟨w,xi⟩Z\_\{i\}=\\langle w,x\_\{i\}\\rangle\. ThenZiZ\_\{i\}satisfies the\(k,vw,p\)\(k,v\_\{w\},p\), wherevw=w⊤​Γs​b​w,p=320v\_\{w\}=\\sqrt\{w^\{\\top\}\\Gamma\_\{sb\}w\},p=\\frac\{3\}\{20\}\. Proposition 2\.5 says that If\(Zi\)i=1T\(Z\_\{i\}\)\_\{i=1\}^\{T\}satisfies\(k,ν,p\)\(k,\\nu,p\)\-BMSB, then

Pr\(∑i=1TZi2≤ν2​p28k⌊T/k⌋\)≤exp\(−⌊T/k⌋p2/8\)\.\\Pr\\\!\\left\(\\sum\_\{i=1\}^\{T\}Z\_\{i\}^\{2\}\\leq\\frac\{\\nu^\{2\}p^\{2\}\}\{8\}\\,k\\lfloor T/k\\rfloor\\right\)\\leq\\exp\\\!\\left\(\-\\lfloor T/k\\rfloor p^\{2\}/8\\right\)\.Applying this withT=tT=tandν=νw\\nu=\\nu\_\{w\},

Pr\(w⊤XtXt⊤w≤p28k⌊t/k⌋w⊤Γsbw\)≤exp\(−⌊t/k⌋p2/8\)\.\\Pr\\\!\\left\(w^\{\\top\}X\_\{t\}X\_\{t\}^\{\\top\}w\\leq\\frac\{p^\{2\}\}\{8\}\\,k\\lfloor t/k\\rfloor\\,w^\{\\top\}\\Gamma\_\{\\mathrm\{sb\}\}w\\right\)\\leq\\exp\\\!\\left\(\-\\lfloor t/k\\rfloor p^\{2\}/8\\right\)\.By Lemma 4\.1\[pmlr\-v75\-simchowitz18a\]\( in our notationQ=Xt⊤Q=X\_\{t\}^\{\\top\}\) , ifinfw∈𝒯w⊤​Xt​Xt⊤​w≥1\\inf\_\{w\\in\\mathcal\{T\}\}w^\{\\top\}X\_\{t\}X\_\{t\}^\{\\top\}w\\geq 1andXt​Xt⊤≲ΓmaxX\_\{t\}X\_\{t\}^\{\\top\}\\lesssim\\Gamma\_\{\\mathrm\{max\}\}, then

Xt​Xt⊤⪰Γmin2=p216​k​⌊t/k⌋​Γsb\.X\_\{t\}X\_\{t\}^\{\\top\}\\succeq\\frac\{\\Gamma\_\{\\min\}\}\{2\}=\\frac\{p^\{2\}\}\{16\}\\,k\\lfloor t/k\\rfloor\\Gamma\_\{\\mathrm\{sb\}\}\.\(75\)Define the following events \(Γ¯=∑m=0t−1Gm​Gm⊤\\bar\{\\Gamma\}=\\sum\_\{m=0\}^\{t\-1\}G\_\{m\}G\_\{m\}^\{\\top\}\)

ℰ1:=\{∥WtV∥op≥K\},ℰ2:=\{XtXt⊤⪰k⁡⌊t/k⌋​p2​Γsb16\},ℰ3:=\{XtXt⊤⋠tΓ¯\}\.\\mathcal\{E\}\_\{1\}:=\\left\\\{\\\|W\_\{t\}V\\\|\_\{\\mathrm\{op\}\}\\geq K\\right\\\},\\qquad\\mathcal\{E\}\_\{2\}:=\\left\\\{X\_\{t\}X\_\{t\}^\{\\top\}\\succeq\\frac\{k\\lfloor t/k\\rfloor p^\{2\}\\Gamma\_\{\\mathrm\{sb\}\}\}\{16\}\\right\\\},\\qquad\\mathcal\{E\}\_\{3\}:=\\left\\\{X\_\{t\}X\_\{t\}^\{\\top\}\\npreceq t\\bar\{\\Gamma\}\\right\\\}\.ℙ\[\{∥WtXt⊤\(XtXt⊤\)−1∥op≥4​Kp​k⁡⌊t/k⌋​λmin​\(Γsb\)\}\]≤ℙ\[ℰ1∩ℰ2∩ℰ3c\]\+ℙ\[ℰ2c∩ℰ3c\]\+ℙ\[ℰ3\]\.\\displaystyle\\mathbb\{P\}\\\!\\left\[\\left\\\{\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\geq\\frac\{4K\}\{p\\sqrt\{k\\lfloor t/k\\rfloor\\,\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\}\\right\\\}\\right\]\\leq\\mathbb\{P\}\[\\mathcal\{E\}\_\{1\}\\cap\\mathcal\{E\}\_\{2\}\\cap\\mathcal\{E\}\_\{3\}^\{c\}\]\+\\mathbb\{P\}\[\\mathcal\{E\}\_\{2\}^\{c\}\\cap\\mathcal\{E\}\_\{3\}^\{c\}\]\+\\mathbb\{P\}\[\\mathcal\{E\}\_\{3\}\]\.By the same argument of boundingℙ⁡\[ℰ1∩ℰ2∩ℰ3c\]\\mathbb\{P\}\[\\mathcal\{E\}\_\{1\}\\cap\\mathcal\{E\}\_\{2\}\\cap\\mathcal\{E\}\_\{3\}^\{c\}\]andℙ⁡\[ℰ2c∩ℰ3c\]\\mathbb\{P\}\[\\mathcal\{E\}\_\{2\}^\{c\}\\cap\\mathcal\{E\}\_\{3\}^\{c\}\]and byℙ⁡\[ℰ3\]≤δ\\mathbb\{P\}\[\\mathcal\{E\}\_\{3\}\]\\leq\\delta, we have

ℙ\[∥WtXt⊤\(XtXt⊤\)−1∥op\>90​σpn\+nlog10p\+logdetΓ¯Γsb−1\+log\(1δ\)t​λmin​\(Γsb\)\]≤3δ,\\mathbb\{P\}\\\!\\left\[\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\>\\frac\{90\\sigma\}\{p\}\\sqrt\{\\frac\{n\+n\\log\\frac\{10\}\{p\}\+\\log\\det\\overline\{\\Gamma\}\\Gamma\_\{\\mathrm\{sb\}\}^\{\-1\}\+\\log\\\!\\left\(\\frac\{1\}\{\\delta\}\\right\)\}\{t\\,\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\}\\right\]\\leq 3\\delta,if

t≥10​kp2\(log\(1δ\)\+2dlog\(10/p\)\+logdet\(Γsb−1\)\)\.t\\geq\\frac\{10k\}\{p^\{2\}\}\\left\(\\log\\\!\\left\(\\frac\{1\}\{\\delta\}\\right\)\+2d\\log\(10/p\)\+\\log\\det\(\\Gamma\_\{\\mathrm\{sb\}\}^\{\-1\}\)\\right\)\.Since

ℙ\[XtXt⊤⋠σ2​dδtΓt\]≤δd​σ2𝔼\[tr\(\(tΓt\)−1/2XtXt⊤\(tΓt\)−1/2\)\]≤δ,\\mathbb\{P\}\\\!\\left\[X\_\{t\}X\_\{t\}^\{\\top\}\\npreceq\\frac\{\\sigma^\{2\}d\}\{\\delta\}\\,t\\Gamma\_\{t\}\\right\]\\leq\\frac\{\\delta\}\{d\\sigma^\{2\}\}\\,\\mathbb\{E\}\\\!\\left\[\\operatorname\{tr\}\\\!\\left\(\(t\\Gamma\_\{t\}\)^\{\-1/2\}X\_\{t\}X\_\{t\}^\{\\top\}\(t\\Gamma\_\{t\}\)^\{\-1/2\}\\right\)\\right\]\\leq\\delta,then there exist universal constantsc,C\>0c,C\>0such that

ℙ\[∥WtXt⊤\(XtXt⊤\)−1∥op\>CT​λmin​\(Γk\)dlogdδ\+logdet\(ΓTΓk−1\)\]≤δ,\\mathbb\{P\}\\\!\\left\[\\bigl\\\|W\_\{t\}X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\>\\frac\{C\}\{\\sqrt\{T\\,\\lambda\_\{\\min\}\(\\Gamma\_\{k\}\)\}\}\\sqrt\{d\\log\\frac\{d\}\{\\delta\}\+\\log\\det\(\\Gamma\_\{T\}\\Gamma\_\{k\}^\{\-1\}\)\}\\right\]\\leq\\delta,for anykksuch thatTk≥c\(dlog\(d/δ\)\+logdet\(ΓTΓk−1\)\)\\frac\{T\}\{k\}\\geq c\\\!\\left\(d\\log\(d/\\delta\)\+\\log\\det\(\\Gamma\_\{T\}\\Gamma\_\{k\}^\{\-1\}\)\\right\)holds\. ∎

### C\.2Proof of[Lemma4](https://arxiv.org/html/2609.18127#Thmlemma4)

###### Proof\.

For each rowi∈\[n\]i\\in\[n\]ands=0,…,t−1s=0,\\ldots,t\-1, recall that

bs\(i\)​\(αi\)=\(Δαi−Δαi,⋆\)​xs\+1\(i\)\.b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)=\\bigl\(\\Delta^\{\\alpha\_\{i\}\}\-\\Delta^\{\\alpha\_\{i,\\star\}\}\\bigr\)x\_\{s\+1\}^\{\(i\)\}\.Since thej=0j=0coefficient does not depend onαi\\alpha\_\{i\}, we have

bs\(i\)​\(αi\)\\displaystyle b\_\{s\}^\{\(i\)\}\(\\alpha\_\{i\}\)=∑j=1s\+1\(ψ⁡\(αi,j\)−ψ⁡\(αi,⋆,j\)\)​xs\+1−j\(i\)\\displaystyle=\\sum\_\{j=1\}^\{s\+1\}\\bigl\(\\psi\(\\alpha\_\{i\},j\)\-\\psi\(\\alpha\_\{i,\\star\},j\)\\bigr\)x\_\{s\+1\-j\}^\{\(i\)\}=∫αi,⋆αi∑j=1s\+1∂uψ⁡\(u,j\)​xs\+1−j\(i\)​𝑑u\.\\displaystyle=\\int\_\{\\alpha\_\{i,\\star\}\}^\{\\alpha\_\{i\}\}\\sum\_\{j=1\}^\{s\+1\}\\partial\_\{u\}\\psi\(u,j\)\\,x\_\{s\+1\-j\}^\{\(i\)\}\\,du\.
Let

Bt\(i\)​\(αi\):=\(b0\(i\)​\(αi\),…,bt−1\(i\)​\(αi\)\)∈ℝ1×t,B\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\):=\\bigl\(b\_\{0\}^\{\(i\)\}\(\\alpha\_\{i\}\),\\ldots,b\_\{t\-1\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\bigr\)\\in\\mathbb\{R\}^\{1\\times t\},and

Bt​\(𝜶\):=\[Bt\(1\)​\(α1\)Bt\(n\)​\(αn\)\]∈ℝn×t\.B\_\{t\}\(\\boldsymbol\{\\alpha\}\):=\\begin\{bmatrix\}B\_\{t\}^\{\(1\)\}\(\\alpha\_\{1\}\)\\\\ \\vdots\\\\ B\_\{t\}^\{\(n\)\}\(\\alpha\_\{n\}\)\\end\{bmatrix\}\\in\\mathbb\{R\}^\{n\\times t\}\.
For each rowii, define

S1,i:=supu∈𝒜i∑j≥1\|∂uψ⁡\(u,j\)\|,S1:=maxi∈\[n\]⁡S1,i\.S\_\{1,i\}:=\\sup\_\{u\\in\\mathcal\{A\}\_\{i\}\}\\sum\_\{j\\geq 1\}\|\\partial\_\{u\}\\psi\(u,j\)\|,\\qquad S\_\{1\}:=\\max\_\{i\\in\[n\]\}S\_\{1,i\}\.By Minkowski’s integral inequality and Young’s convolution inequality,

‖Bt\(i\)​\(αi\)‖2\\displaystyle\\\|B\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\\|\_\{2\}≤\|αi−αi,⋆\|​supu∈𝒜i\(∑s=0t−1\|∑j=1s\+1∂uψ⁡\(u,j\)​xs\+1−j\(i\)\|2\)1/2\\displaystyle\\leq\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\sup\_\{u\\in\\mathcal\{A\}\_\{i\}\}\\left\(\\sum\_\{s=0\}^\{t\-1\}\\left\|\\sum\_\{j=1\}^\{s\+1\}\\partial\_\{u\}\\psi\(u,j\)x\_\{s\+1\-j\}^\{\(i\)\}\\right\|^\{2\}\\right\)^\{1/2\}≤\|αi−αi,⋆\|​supu∈𝒜i\(∑j=1t\|∂uψ⁡\(u,j\)\|\)​\(∑s=0t−1\|xs\(i\)\|2\)1/2\\displaystyle\\leq\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\sup\_\{u\\in\\mathcal\{A\}\_\{i\}\}\\left\(\\sum\_\{j=1\}^\{t\}\|\\partial\_\{u\}\\psi\(u,j\)\|\\right\)\\left\(\\sum\_\{s=0\}^\{t\-1\}\|x\_\{s\}^\{\(i\)\}\|^\{2\}\\right\)^\{1/2\}≤S1,i​\|αi−αi,⋆\|​\(∑s=0t−1\|xs\(i\)\|2\)1/2\.\\displaystyle\\leq S\_\{1,i\}\|\\alpha\_\{i\}\-\\alpha\_\{i,\\star\}\|\\left\(\\sum\_\{s=0\}^\{t\-1\}\|x\_\{s\}^\{\(i\)\}\|^\{2\}\\right\)^\{1/2\}\.Consequently,

‖Bt​\(𝜶\)‖F2=∑i=1n‖Bt\(i\)​\(αi\)‖22≤S12​‖𝜶−𝜶⋆‖∞2​∑i=1n∑s=0t−1\|xs\(i\)\|2=S12​‖𝜶−𝜶⋆‖∞2​‖Xt‖F2\.\\displaystyle\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}^\{2\}=\\sum\_\{i=1\}^\{n\}\\\|B\_\{t\}^\{\(i\)\}\(\\alpha\_\{i\}\)\\\|\_\{2\}^\{2\}\\leq S\_\{1\}^\{2\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\sum\_\{i=1\}^\{n\}\\sum\_\{s=0\}^\{t\-1\}\|x\_\{s\}^\{\(i\)\}\|^\{2\}=S\_\{1\}^\{2\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}^\{2\}\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\.
On the eventXt​Xt⊤≻0X\_\{t\}X\_\{t\}^\{\\top\}\\succ 0,

‖Bt​\(𝜶\)​Xt⊤​\(Xt​Xt⊤\)−1‖op\\displaystyle\\bigl\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}≤‖Bt​\(𝜶\)‖F​‖Xt⊤​\(Xt​Xt⊤\)−1‖op\\displaystyle\\leq\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}\\bigl\\\|X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}=‖Bt​\(𝜶\)‖Fσmin​\(Xt\)\\displaystyle=\\frac\{\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)\\\|\_\{F\}\}\{\\sigma\_\{\\min\}\(X\_\{t\}\)\}≤S1​‖𝜶−𝜶⋆‖∞​‖Xt‖Fσmin​\(Xt\)\.\\displaystyle\\leq S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\frac\{\\\|X\_\{t\}\\\|\_\{F\}\}\{\\sigma\_\{\\min\}\(X\_\{t\}\)\}\.\(76\)
It remains to control‖Xt‖F/σmin​\(Xt\)\\\|X\_\{t\}\\\|\_\{F\}/\\sigma\_\{\\min\}\(X\_\{t\}\)\. Sincex0=0x\_\{0\}=0, the finite\-past representation gives, fors≥1s\\geq 1,

xs=∑m=0s−1Gm​ηs−1−m\.x\_\{s\}=\\sum\_\{m=0\}^\{s\-1\}G\_\{m\}\\eta\_\{s\-1\-m\}\.Therefore,

𝔼​‖Xt‖F2=∑s=0t−1𝔼​‖xs‖22=σ2​∑s=1t−1∑m=0s−1‖Gm‖F2≤t​σ2​∑m=0∞‖Gm‖F2\.\\displaystyle\\mathbb\{E\}\\\|X\_\{t\}\\\|\_\{F\}^\{2\}=\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\\|x\_\{s\}\\\|\_\{2\}^\{2\}=\\sigma^\{2\}\\sum\_\{s=1\}^\{t\-1\}\\sum\_\{m=0\}^\{s\-1\}\\\|G\_\{m\}\\\|\_\{F\}^\{2\}\\leq t\\sigma^\{2\}\\sum\_\{m=0\}^\{\\infty\}\\\|G\_\{m\}\\\|\_\{F\}^\{2\}\.Let

𝒢2:=∑m=0∞‖Gm‖F2\.\\mathcal\{G\}\_\{2\}:=\\sum\_\{m=0\}^\{\\infty\}\\\|G\_\{m\}\\\|\_\{F\}^\{2\}\.By Markov’s inequality, with probability at least1−δ1\-\\delta,

‖Xt‖F2≤t​σ2​𝒢2δ\.\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\\leq\\frac\{t\\sigma^\{2\}\\mathcal\{G\}\_\{2\}\}\{\\delta\}\.\(77\)
Moreover, by the block small\-ball lower bound, with probability at least1−δ1\-\\delta,

Xt​Xt⊤⪰p216​k​⌊t/k⌋​Γsb\.X\_\{t\}X\_\{t\}^\{\\top\}\\succeq\\frac\{p^\{2\}\}\{16\}k\\lfloor t/k\\rfloor\\Gamma\_\{\\mathrm\{sb\}\}\.\(78\)Hence, on the intersection of \([77](https://arxiv.org/html/2609.18127#A3.E77)\) and \([78](https://arxiv.org/html/2609.18127#A3.E78)\),

‖Xt‖F2σmin2​\(Xt\)=‖Xt‖F2λmin​\(Xt​Xt⊤\)≤16​t​σ2​𝒢2δ​p2​k​⌊t/k⌋​λmin​\(Γsb\)≤16​t​σ2​𝒢2δ​p2​\(t−k\)​λmin​\(Γsb\)\.\\displaystyle\\frac\{\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\}\{\\sigma\_\{\\min\}^\{2\}\(X\_\{t\}\)\}=\\frac\{\\\|X\_\{t\}\\\|\_\{F\}^\{2\}\}\{\\lambda\_\{\\min\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)\}\\leq\\frac\{16t\\sigma^\{2\}\\mathcal\{G\}\_\{2\}\}\{\\delta p^\{2\}k\\lfloor t/k\\rfloor\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\\leq\\frac\{16t\\sigma^\{2\}\\mathcal\{G\}\_\{2\}\}\{\\delta p^\{2\}\(t\-k\)\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\.Ift≥10​kt\\geq 10k, thent/\(t−k\)≤10/9t/\(t\-k\)\\leq 10/9, and thus

‖Xt‖Fσmin​\(Xt\)≤160​σ2​𝒢29​δ​p2​λmin​\(Γsb\)\.\\frac\{\\\|X\_\{t\}\\\|\_\{F\}\}\{\\sigma\_\{\\min\}\(X\_\{t\}\)\}\\leq\\sqrt\{\\frac\{160\\sigma^\{2\}\\mathcal\{G\}\_\{2\}\}\{9\\delta p^\{2\}\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\}\.\(79\)
Combining \([76](https://arxiv.org/html/2609.18127#A3.E76)\) and \([79](https://arxiv.org/html/2609.18127#A3.E79)\), we obtain, with probability at least1−2​δ1\-2\\delta,

‖Bt​\(𝜶\)​Xt⊤​\(Xt​Xt⊤\)−1‖op≤S1​‖𝜶−𝜶⋆‖∞​160​σ2​𝒢29​δ​p2​λmin​\(Γsb\)\.\\bigl\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\leq S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\sqrt\{\\frac\{160\\sigma^\{2\}\\mathcal\{G\}\_\{2\}\}\{9\\delta p^\{2\}\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\}\.
Finally, Lemma[14](https://arxiv.org/html/2609.18127#Thmlemma14)gives

𝒢2≤n​C~G2​∑m=0∞\(m\+1\)−2​\(1\+αmin\)=n​C~G2​ζ​\(2\+2​αmin\)≤2​n​C~G2,\\mathcal\{G\}\_\{2\}\\leq n\\widetilde\{C\}\_\{G\}^\{2\}\\sum\_\{m=0\}^\{\\infty\}\(m\+1\)^\{\-2\(1\+\\alpha\_\{\\min\}\)\}=n\\tilde\{C\}\_\{G\}^\{2\}\\zeta\(2\+2\\alpha\_\{\\min\}\)\\leq 2n\\tilde\{C\}\_\{G\}^\{2\},and therefore

‖Bt​\(𝜶\)​Xt⊤​\(Xt​Xt⊤\)−1‖op≤S1​‖𝜶−𝜶⋆‖∞​320​n​σ2​C~G29​δ​p2​λmin​\(Γsb\)\.\\bigl\\\|B\_\{t\}\(\\boldsymbol\{\\alpha\}\)X\_\{t\}^\{\\top\}\(X\_\{t\}X\_\{t\}^\{\\top\}\)^\{\-1\}\\bigr\\\|\_\{\\mathrm\{op\}\}\\leq S\_\{1\}\\\|\\boldsymbol\{\\alpha\}\-\\boldsymbol\{\\alpha\}\_\{\\star\}\\\|\_\{\\infty\}\\sqrt\{\\frac\{320n\\sigma^\{2\}\\tilde\{C\}\_\{G\}^\{2\}\}\{9\\delta p^\{2\}\\lambda\_\{\\min\}\(\\Gamma\_\{\\mathrm\{sb\}\}\)\}\}\.\(80\)This proves the claim\. ∎

## Appendix DAuxiliary Lemmas

###### Lemma 14\.

Suppose Assumption 1 holds and letαmin:=mini∈\[n\]⁡αi,⋆\.\\alpha\_\{\\min\}:=\\min\_\{i\\in\[n\]\}\\alpha\_\{i,\\star\}\.Then there exists a constantC~G<∞\\widetilde\{C\}\_\{G\}<\\infty, depending only on\(A⋆,α⋆\)\(A\_\{\\star\},\\alpha\_\{\\star\}\), such that

‖Gm‖op≤C~G​\(m\+1\)−\(1\+αmin\),m≥0\.\\\|G\_\{m\}\\\|\_\{\\mathrm\{op\}\}\\leq\\widetilde\{C\}\_\{G\}\(m\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\},\\qquad m\\geq 0\.\(81\)Consequently, for everyp≥1p\\geq 1andk≥0k\\geq 0,

‖Cov⁡\(xp,xp\+k\)‖op≤C~x​\(k\+1\)−\(1\+αmin\),\\\|\\operatorname\{Cov\}\(x\_\{p\},x\_\{p\+k\}\)\\\|\_\{\\mathrm\{op\}\}\\leq\\widetilde\{C\}\_\{x\}\(k\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\},\(82\)where

C~x:=σ2​C~G2​\(1\+1αmin\+12​αmin\+1\)\.\\widetilde\{C\}\_\{x\}:=\\sigma^\{2\}\\widetilde\{C\}\_\{G\}^\{2\}\\left\(1\+\\frac\{1\}\{\\alpha\_\{\\min\}\}\+\\frac\{1\}\{2\\alpha\_\{\\min\}\+1\}\\right\)\.

###### Proof\.

we first boundGmG\_\{m\}\. LetΨj:=diag⁡\(ψ⁡\(α1,⋆,j\),…,ψ⁡\(αn,⋆,j\)\)\.\\Psi\_\{j\}:=\\operatorname\{diag\}\\bigl\(\\psi\(\\alpha\_\{1,\\star\},j\),\\ldots,\\psi\(\\alpha\_\{n,\\star\},j\)\\bigr\)\.We have

𝒜⁡\(z\)=A⋆\+diag⁡\(𝜶⋆\)−∑j=1∞Ψj\+1​zj\.\\mathcal\{A\}\(z\)=A\_\{\\star\}\+\\operatorname\{diag\}\(\\boldsymbol\{\\alpha\}\_\{\\star\}\)\-\\sum\_\{j=1\}^\{\\infty\}\\Psi\_\{j\+1\}z^\{j\}\.For each coordinateii,∑k=0∞ψ⁡\(αi,⋆,k\)​zk=\(1−z\)αi,⋆\.\\sum\_\{k=0\}^\{\\infty\}\\psi\(\\alpha\_\{i,\\star\},k\)\\,z^\{k\}=\(1\-z\)^\{\\alpha\_\{i,\\star\}\}\.Therefore

∑j=1∞ψ⁡\(αi,⋆,j\+1\)​zj=\(1−z\)αi,⋆−1\+αi,⋆​zz\.\\sum\_\{j=1\}^\{\\infty\}\\psi\(\\alpha\_\{i,\\star\},j\+1\)\\,z^\{j\}=\\frac\{\(1\-z\)^\{\\alpha\_\{i,\\star\}\}\-1\+\\alpha\_\{i,\\star\}z\}\{z\}\.DefineD𝜶⋆​\(z\):=diag⁡\(\(1−z\)α1,⋆,…,\(1−z\)αn,⋆\)\.D\_\{\\boldsymbol\{\\alpha\}\_\{\\star\}\}\(z\):=\\operatorname\{diag\}\\bigl\(\(1\-z\)^\{\\alpha\_\{1,\\star\}\},\\ldots,\(1\-z\)^\{\\alpha\_\{n,\\star\}\}\\bigr\)\.Then

𝒢⁡\(z\)=\(D𝜶⋆​\(z\)−z​A⋆\)−1\.\\mathcal\{G\}\(z\)=\\bigl\(D\_\{\\boldsymbol\{\\alpha\}\_\{\\star\}\}\(z\)\-zA\_\{\\star\}\\bigr\)^\{\-1\}\.Letw=1−zw=1\-zandD⁡\(w\):=diag⁡\(wα1,⋆,…,wαn,⋆\)\.D\(w\):=\\operatorname\{diag\}\\bigl\(w^\{\\alpha\_\{1,\\star\}\},\\ldots,w^\{\\alpha\_\{n,\\star\}\}\\bigr\)\.Then

𝒢⁡\(z\)=−\(I−B⁡\(w\)\)−1​A⋆−1,\\mathcal\{G\}\(z\)=\-\\,\\bigl\(I\-B\(w\)\\bigr\)^\{\-1\}A\_\{\\star\}^\{\-1\},whereB⁡\(w\):=w​I\+A⋆−1​D​\(w\)B\(w\):=wI\+A\_\{\\star\}^\{\-1\}D\(w\)\. Define

R⁡\(z\):=B​\(w\)2​\(I−B⁡\(w\)\)−1​A⋆−1\.R\(z\):=B\(w\)^\{2\}\(I\-B\(w\)\)^\{\-1\}A\_\{\\star\}^\{\-1\}\.Then, we have

Gm=−A⋆−1​diag⁡\(ψ⁡\(α1,⋆,m\),…,ψ⁡\(αn,⋆,m\)\)​A⋆−1−\[zm\]​R​\(z\),m≥2\.G\_\{m\}=\-\\,A\_\{\\star\}^\{\-1\}\\operatorname\{diag\}\\bigl\(\\psi\(\\alpha\_\{1,\\star\},m\),\\ldots,\\psi\(\\alpha\_\{n,\\star\},m\)\\bigr\)A\_\{\\star\}^\{\-1\}\-\[z^\{m\}\]R\(z\),\\quad m\\geq 2\.For\|w\|≤1\|w\|\\leq 1,

‖D⁡\(w\)‖op=maxi⁡\|w\|αi,⋆≤\|w\|αmin,\\\|D\(w\)\\\|\_\{\\mathrm\{op\}\}=\\max\_\{i\}\|w\|^\{\\alpha\_\{i,\\star\}\}\\leq\|w\|^\{\\alpha\_\{\\min\}\},and therefore

‖B⁡\(w\)‖op≤\(1\+‖A⋆−1‖op\)​\|w\|αmin\.\\\|B\(w\)\\\|\_\{\\mathrm\{op\}\}\\leq\\bigl\(1\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\bigr\)\|w\|^\{\\alpha\_\{\\min\}\}\.Chooseρ\>0\\rho\>0sufficiently small so that‖B⁡\(w\)‖op≤1/2\\\|B\(w\)\\\|\_\{\\mathrm\{op\}\}\\leq 1/2whenever\|w\|≤ρ\|w\|\\leq\\rho\. Then

‖\(I−B⁡\(w\)\)−1‖op≤2,\\\|\(I\-B\(w\)\)^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\leq 2,and thus

‖R⁡\(z\)‖op≤2​‖A⋆−1‖op​‖B⁡\(w\)‖op2≤2​‖A⋆−1‖op​\(1\+‖A⋆−1‖op\)2​\|w\|2​αmin\.\\\|R\(z\)\\\|\_\{\\mathrm\{op\}\}\\leq 2\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\\|B\(w\)\\\|\_\{\\mathrm\{op\}\}^\{2\}\\leq 2\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\bigl\(1\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\bigr\)^\{2\}\|w\|^\{2\\alpha\_\{\\min\}\}\.Using the singularity transfer argument in[Lemma15](https://arxiv.org/html/2609.18127#Thmlemma15), we have

‖\[zm\]​R​\(z\)‖op≤CI​‖A⋆−1‖op​\(1\+‖A⋆−1‖op\)2​m−\(1\+2​αmin\),\\\|\[z^\{m\}\]R\(z\)\\\|\_\{\\mathrm\{op\}\}\\leq C\_\{I\}\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\bigl\(1\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\\bigr\)^\{2\}\\,m^\{\-\(1\+2\\alpha\_\{\\min\}\)\},whereCIC\_\{I\}depends only on the interval forαi,⋆\\alpha\_\{i,\\star\}\.

We next bound the fractional coefficientψ⁡\(α,j\)\\psi\(\\alpha,j\)\. For0<α<10<\\alpha<1,

ψ⁡\(α,j\)=Γ⁡\(j−α\)Γ⁡\(−α\)​Γ​\(j\+1\)=\(−1\)j​\(αj\)\.\\psi\(\\alpha,j\)=\\frac\{\\Gamma\(j\-\\alpha\)\}\{\\Gamma\(\-\\alpha\)\\Gamma\(j\+1\)\}=\(\-1\)^\{j\}\\binom\{\\alpha\}\{j\}\.Forj≥2j\\geq 2,

\|ψ⁡\(α,j\)\|=αj​∏k=1j−1\(1−αk\)\.\|\\psi\(\\alpha,j\)\|=\\frac\{\\alpha\}\{j\}\\prod\_\{k=1\}^\{j\-1\}\\left\(1\-\\frac\{\\alpha\}\{k\}\\right\)\.Usinglog⁡\(1−u\)≤−u\\log\(1\-u\)\\leq\-ufor0<u<10<u<1and∑k=1j−1k−1≥log⁡j\\sum\_\{k=1\}^\{j\-1\}k^\{\-1\}\\geq\\log j, we obtain

\|ψ⁡\(α,j\)\|≤α​j−\(1\+α\),j≥1\.\|\\psi\(\\alpha,j\)\|\\leq\\alpha j^\{\-\(1\+\\alpha\)\},\\qquad j\\geq 1\.Forα=1\\alpha=1, the same bound follows directly fromψ⁡\(1,1\)=−1\\psi\(1,1\)=\-1andψ⁡\(1,j\)=0\\psi\(1,j\)=0forj≥2j\\geq 2\.

Hence

maxi∈\[n\]⁡\|ψ⁡\(αi,⋆,m\)\|≤m−\(1\+αmin\),m≥1\.\\max\_\{i\\in\[n\]\}\|\\psi\(\\alpha\_\{i,\\star\},m\)\|\\leq m^\{\-\(1\+\\alpha\_\{\\min\}\)\},\\qquad m\\geq 1\.Then, we have

‖Gm‖op≤‖A⋆−2‖op​m−\(1\+αmin\)\+CI​‖A⋆−1‖op​\(1\+‖A⋆−1‖op\)2​m−\(1\+2​αmin\)\.\\\|G\_\{m\}\\\|\_\{\\mathrm\{op\}\}\\leq\\\|A\_\{\\star\}^\{\-2\}\\\|\_\{\\mathrm\{op\}\}\\,m^\{\-\(1\+\\alpha\_\{\\min\}\)\}\+C\_\{I\}\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\(1\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\)^\{2\}m^\{\-\(1\+2\\alpha\_\{\\min\}\)\}\.Thus

‖Gm‖op≤C~G​\(m\+1\)−\(1\+αmin\),m≥0,\\\|G\_\{m\}\\\|\_\{\\mathrm\{op\}\}\\leq\\tilde\{C\}\_\{G\}\(m\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\},\\;m\\geq 0,\(83\)whereC~G=CI′​\(‖A⋆−2‖op\+‖A⋆−1‖op​\(1\+‖A⋆−1‖op\)2\)\\tilde\{C\}\_\{G\}=C\_\{I\}^\{\\prime\}\\Bigl\(\\\|A\_\{\\star\}^\{\-2\}\\\|\_\{\\mathrm\{op\}\}\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\(1\+\\\|A\_\{\\star\}^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\)^\{2\}\\Bigr\)\.

Therefore,

‖Cov⁡\(xp,xp\+k\)‖op\\displaystyle\\\|\\operatorname\{Cov\}\(x\_\{p\},x\_\{p\+k\}\)\\\|\_\{\\mathrm\{op\}\}≤σ2​∑m=0p−1‖Gm‖op​‖Gm\+k‖op\\displaystyle\\leq\\sigma^\{2\}\\sum\_\{m=0\}^\{p\-1\}\\\|G\_\{m\}\\\|\_\{\\mathrm\{op\}\}\\\|G\_\{m\+k\}\\\|\_\{\\mathrm\{op\}\}≤σ2​C~G2​∑m=0∞\(m\+1\)−\(1\+αmin\)​\(m\+k\+1\)−\(1\+αmin\)\.\\displaystyle\\leq\\sigma^\{2\}\\widetilde\{C\}\_\{G\}^\{2\}\\sum\_\{m=0\}^\{\\infty\}\(m\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\(m\+k\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\.Splitting the last sum into0≤m≤k0\\leq m\\leq kandm\>km\>kgives

∑m=0∞\(m\+1\)−\(1\+αmin\)​\(m\+k\+1\)−\(1\+αmin\)≤\(1\+1αmin\+12​αmin\+1\)​\(k\+1\)−\(1\+αmin\)\.\\sum\_\{m=0\}^\{\\infty\}\(m\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\(m\+k\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\\leq\\left\(1\+\\frac\{1\}\{\\alpha\_\{\\min\}\}\+\\frac\{1\}\{2\\alpha\_\{\\min\}\+1\}\\right\)\(k\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\.Thus

‖Cov⁡\(xp,xp\+k\)‖op≤C~x​\(k\+1\)−\(1\+αmin\)\.\\\|\\operatorname\{Cov\}\(x\_\{p\},x\_\{p\+k\}\)\\\|\_\{\\mathrm\{op\}\}\\leq\\widetilde\{C\}\_\{x\}\(k\+1\)^\{\-\(1\+\\alpha\_\{\\min\}\)\}\.The caseq<pq<pfollows by covariance symmetry, which proves the claim for allp,q≥0p,q\\geq 0\. ∎

###### Lemma 15\.

Fixϕ∈\(0,π/2\)\\phi\\in\(0,\\pi/2\)andβ0\>0\\beta\_\{0\}\>0\. Then there exists a constantKϕ,β0<∞K\_\{\\phi,\\beta\_\{0\}\}<\\inftysuch that the following holds\.

Letρ∈\(0,1\]\\rho\\in\(0,1\], and let

Δ\(ϕ,ρ\):=\{z:\|z\|<1\+ρ,z≠1,\|arg\(z−1\)\|\>ϕ\}\.\\Delta\(\\phi,\\rho\):=\\\{z:\|z\|<1\+\\rho,\\ z\\neq 1,\\ \|\\arg\(z\-1\)\|\>\\phi\\\}\.Ifffis analytic inΔ⁡\(ϕ,ρ\)\\Delta\(\\phi,\\rho\)and satisfies

\|f⁡\(z\)\|≤M​\|1−z\|β,z∈Δ⁡\(ϕ,ρ\),\|f\(z\)\|\\leq M\|1\-z\|^\{\\beta\},\\qquad z\\in\\Delta\(\\phi,\\rho\),for someβ∈\[0,β0\]\\beta\\in\[0,\\beta\_\{0\}\], then for everym≥1m\\geq 1,

\|\[zm\]​f​\(z\)\|≤Kϕ,β0​M​ρ−\(β\+1\)​m−\(β\+1\)\.\|\[z^\{m\}\]f\(z\)\|\\leq K\_\{\\phi,\\beta\_\{0\}\}\\,M\\,\\rho^\{\-\(\\beta\+1\)\}\\,m^\{\-\(\\beta\+1\)\}\.

###### Proof\.

We follow the standard truncated Hankel\-contour proof of the Flajolet–Odlyzko transfer theorem\[Flajolet1990SingularityAO\], keeping theρ\\rho\-dependence explicit\.

Form<2​ρ−1m<2\\rho^\{\-1\}, the right\-hand side is≥2−\(β\+1\)≥2−\(β0\+1\)\\geq 2^\{\-\(\\beta\+1\)\}\\geq 2^\{\-\(\\beta\_\{0\}\+1\)\}\. Sinceffis analytic on a fixed contour insideΔ⁡\(ϕ,ρ\)\\Delta\(\\phi,\\rho\)enclosing the origin, Cauchy’s formula gives\|\[zm\]​f​\(z\)\|≤Cϕ,β0​M\|\[z^\{m\}\]f\(z\)\|\\leq C\_\{\\phi,\\beta\_\{0\}\}M, so the claim follows after enlargingKϕ,β0K\_\{\\phi,\\beta\_\{0\}\}\. One choice isCϕ,β0=2β0C\_\{\\phi,\\beta\_\{0\}\}=2^\{\\beta\_\{0\}\}andKϕ,β0≥22​β0\+1K\_\{\\phi,\\beta\_\{0\}\}\\geq 2^\{2\\beta\_\{0\}\+1\}\.

Hence, it suffices to considerm≥2​ρ−1m\\geq 2\\rho^\{\-1\}\. Fixϑ=ϕ\+π/22∈\(ϕ,π/2\)\\vartheta=\\frac\{\\phi\+\\pi/2\}\{2\}\\in\(\\phi,\\pi/2\)and letrρ,ϑ\>0r\_\{\\rho,\\vartheta\}\>0be the unique solution to

\|1\+r​ei​ϑ\|=1\+ρ2,\|1\+re^\{i\\vartheta\}\|=1\+\\frac\{\\rho\}\{2\},namely,rρ,ϑ=−cos⁡ϑ\+cos2⁡ϑ\+ρ\+ρ24r\_\{\\rho,\\vartheta\}=\-\\cos\\vartheta\+\\sqrt\{\\cos^\{2\}\\vartheta\+\\rho\+\\frac\{\\rho^\{2\}\}\{4\}\}\. Consider the contourHm,ρ,ϑ∈Δ⁡\(ϕ,ρ\)H\_\{m,\\rho,\\vartheta\}\\in\\Delta\(\\phi,\\rho\)defined by

Γ±:=\{1\+r​e±i​ϑ:m−1≤r≤rρ,ϑ\},\\Gamma\_\{\\pm\}:=\\\{1\+re^\{\\pm i\\vartheta\}:m^\{\-1\}\\leq r\\leq r\_\{\\rho,\\vartheta\}\\\},Γ0:=\{1\+m−1​ei​θ:ϑ≤θ≤2​π−ϑ\},\\Gamma\_\{0\}:=\\\{1\+m^\{\-1\}e^\{i\\theta\}:\\vartheta\\leq\\theta\\leq 2\\pi\-\\vartheta\\\},and the outer arcΓout⊂\{\|z\|=1\+ρ/2\}∩Δ\(ϕ,ρ\)\\Gamma\_\{\\rm out\}\\subset\\\{\|z\|=1\+\\rho/2\\\}\\cap\\Delta\(\\phi,\\rho\)joining the endpoints ofΓ\+\\Gamma\_\{\+\}andΓ−\\Gamma\_\{\-\}\. Therefore,\[zm\]​f​\(z\)=12​π​i​∫Hm,ρ,ϑf⁡\(z\)​z−m−1​𝑑z\.\[z^\{m\}\]f\(z\)=\\frac\{1\}\{2\\pi i\}\\int\_\{H\_\{m,\\rho,\\vartheta\}\}f\(z\)z^\{\-m\-1\}dz\.OnΓ±\\Gamma\_\{\\pm\},\|1−z\|=r,\|d​z\|=d​r\|1\-z\|=r,\|dz\|=dr, and

\|z\|=\|1\+r​e±i​ϑ\|≥1\+r​cos⁡ϑ\.\|z\|=\|1\+re^\{\\pm i\\vartheta\}\|\\geq 1\+r\\cos\\vartheta\.Therefore, onΓ±\\Gamma\_\{\\pm\},\|z\|−m≤\(1\+r​cos⁡ϑ\)−m=e−m​log⁡\(1\+r​cos⁡ϑ\)≤e−m​r​cos⁡ϑ1\+r1,ϑ​cos⁡ϑ=e−cϕ​m​r\|z\|^\{\-m\}\\leq\(1\+r\\cos\\vartheta\)^\{\-m\}=e^\{\-m\\log\(1\+r\\cos\\vartheta\)\}\\leq e^\{\-\\frac\{mr\\cos\\vartheta\}\{1\+r\_\{1,\\vartheta\}\\cos\\vartheta\}\}=e^\{\-c\_\{\\phi\}mr\}, wherecϕ:=cos⁡ϑ1\+r1,ϑ​cos⁡ϑc\_\{\\phi\}:=\\frac\{\\cos\\vartheta\}\{1\+r\_\{1,\\vartheta\}\\cos\\vartheta\}\. Using\|f⁡\(z\)\|≤M​rβ\|f\(z\)\|\\leq Mr^\{\\beta\}there, we obtain

∫Γ±\|f⁡\(z\)​z−m−1​𝑑z\|≤M​∫m−1rρ,ϑrβ​e−cϕ​m​r​𝑑r≤M​\(cϕ​m\)−\(β\+1\)​Γ​\(β\+1\)≤Cϕ,β0​M​m−\(β\+1\)\.\\displaystyle\\int\_\{\\Gamma\_\{\\pm\}\}\|f\(z\)z^\{\-m\-1\}\\,dz\|\\leq M\\int\_\{m^\{\-1\}\}^\{r\_\{\\rho,\\vartheta\}\}r^\{\\beta\}e^\{\-c\_\{\\phi\}mr\}\\,dr\\leq M\(c\_\{\\phi\}m\)^\{\-\(\\beta\+1\)\}\\Gamma\(\\beta\+1\)\\leq C\_\{\\phi,\\beta\_\{0\}\}Mm^\{\-\(\\beta\+1\)\}\.
OnΓ0\\Gamma\_\{0\}, we have\|1−z\|=m−1\|1\-z\|=m^\{\-1\}and\|f⁡\(z\)\|≤M​m−β\|f\(z\)\|\\leq Mm^\{\-\\beta\}\. The arc length satisfies\(2​π−2​ϑ\)​m−1≤Cϕ​m−1\(2\\pi\-2\\vartheta\)m^\{\-1\}\\leq C\_\{\\phi\}m^\{\-1\}\. Since the smallest possible value of\|z\|\|z\|on the arc occurs atθ=π\\theta=\\pi, we have\|z\|≥1−1m\|z\|\\geq 1\-\\frac\{1\}\{m\}\. Sincem≥2​ρ−1≥2m\\geq 2\\rho^\{\-1\}\\geq 2, we can uniformly upper bound\|z\|−m−1\|z\|^\{\-m\-1\}by\|z\|−m−1≤\(1−1m\)−\(m\+1\)≤8\|z\|^\{\-m\-1\}\\leq\(1\-\\frac\{1\}\{m\}\)^\{\-\(m\+1\)\}\\leq 8\. Hence

∫Γ0\|f⁡\(z\)​z−m−1​𝑑z\|≤8​M​m−β​Cϕ​m−1=Cϕ′​M​m−\(β\+1\)\.\\displaystyle\\int\_\{\\Gamma\_\{0\}\}\|f\(z\)z^\{\-m\-1\}\\,dz\|\\leq 8Mm^\{\-\\beta\}C\_\{\\phi\}m^\{\-1\}=C\_\{\\phi\}^\{\\prime\}Mm^\{\-\(\\beta\+1\)\}\.On the outer arcΓout\\Gamma\_\{\\rm out\},\|z\|=1\+ρ/2\|z\|=1\+\\rho/2, so

\|z\|−m=\(1\+ρ/2\)−m≤e−c​ρ​m\|z\|^\{\-m\}=\(1\+\\rho/2\)^\{\-m\}\\leq e^\{\-c\\rho m\}for a universalc\>0c\>0\. Also, sinceβ≤β0\\beta\\leq\\beta\_\{0\}and\|1−z\|≤3\|1\-z\|\\leq 3on\|z\|=1\+ρ/2\|z\|=1\+\\rho/2withρ≤1\\rho\\leq 1, we have\|f⁡\(z\)\|≤3β0​M\|f\(z\)\|\\leq 3^\{\\beta\_\{0\}\}M\. Hence

∫Γout\|f⁡\(z\)​z−m−1​𝑑z\|≤Cϕ,β0​M​e−c​ρ​m\.\\int\_\{\\Gamma\_\{\\rm out\}\}\|f\(z\)z^\{\-m\-1\}\\,dz\|\\leq C\_\{\\phi,\\beta\_\{0\}\}Me^\{\-c\\rho m\}\.Finally, because

supx\>0xβ0\+1​e−c​x<∞\\sup\_\{x\>0\}x^\{\\beta\_\{0\}\+1\}e^\{\-cx\}<\\inftyandρ≤1\\rho\\leq 1, we have

e−c​ρ​m≤Cβ0​ρ−\(β\+1\)​m−\(β\+1\)\.e^\{\-c\\rho m\}\\leq C\_\{\\beta\_\{0\}\}\\rho^\{\-\(\\beta\+1\)\}m^\{\-\(\\beta\+1\)\}\.Combining the three contour bounds with Cauchy’s coefficient formula proves the claim\.

∎

###### Lemma 16\.

Suppose[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1)holds and consider the stationary two\-sided solution of the system\. Define

sG:=inf\|z\|=1σmin​\(G⁡\(z\)\),αmax:=maxi∈\[n\]⁡αi,⋆\.s\_\{G\}:=\\inf\_\{\|z\|=1\}\\sigma\_\{\\min\}\(G\(z\)\),\\qquad\\alpha\_\{\\max\}:=\\max\_\{i\\in\[n\]\}\\alpha\_\{i,\\star\}\.Then, for everyi∈\[n\]i\\in\[n\],

μt,i\\displaystyle\\mu\_\{t,i\}≥σ2​sG2​\[\(αi,⋆−12\)2\+\(−3​αi,⋆2\+6​αi,⋆−26\)2\]\\displaystyle\\geq\\sigma^\{2\}s\_\{G\}^\{2\}\\left\[\\left\(\\alpha\_\{i,\\star\}\-\\frac\{1\}\{2\}\\right\)^\{2\}\+\\left\(\\frac\{\-3\\alpha\_\{i,\\star\}^\{2\}\+6\\alpha\_\{i,\\star\}\-2\}\{6\}\\right\)^\{2\}\\right\]≥σ2\(2αmax\+‖A⋆‖op\)2​\[\(αi,⋆−12\)2\+\(−3​αi,⋆2\+6​αi,⋆−26\)2\]\.\\displaystyle\\geq\\frac\{\\sigma^\{2\}\}\{\\bigl\(2^\{\\alpha\_\{\\max\}\}\+\\\|A\_\{\\star\}\\\|\_\{\\mathrm\{op\}\}\\bigr\)^\{2\}\}\\left\[\\left\(\\alpha\_\{i,\\star\}\-\\frac\{1\}\{2\}\\right\)^\{2\}\+\\left\(\\frac\{\-3\\alpha\_\{i,\\star\}^\{2\}\+6\\alpha\_\{i,\\star\}\-2\}\{6\}\\right\)^\{2\}\\right\]\.

###### Proof\.

Let

di,j:=∂αψ⁡\(αi,⋆,j\),j≥1,d\_\{i,j\}:=\\partial\_\{\\alpha\}\\psi\(\\alpha\_\{i,\\star\},j\),\\qquad j\\geq 1,and define

gs\(i\):=∑j≥1di,j​xs\+1−j\(i\)\.g\_\{s\}^\{\(i\)\}:=\\sum\_\{j\\geq 1\}d\_\{i,j\}\\,x\_\{s\+1\-j\}^\{\(i\)\}\.By the definition of the population profiled derivative curvature,

μt,i=infai∈ℝ1×n1t​∑s=0t−1𝔼​\|gs\(i\)−ai​xs\|2\.\\mu\_\{t,i\}=\\inf\_\{a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}\\frac\{1\}\{t\}\\sum\_\{s=0\}^\{t\-1\}\\mathbb\{E\}\\left\|g\_\{s\}^\{\(i\)\}\-a\_\{i\}x\_\{s\}\\right\|^\{2\}\.Under stationarity, every term in the preceding average has the same distribution\. Hence

μt,i=infai∈ℝ1×n𝔼​\|g0\(i\)−ai​x0\|2\.\\mu\_\{t,i\}=\\inf\_\{a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}\\mathbb\{E\}\\left\|g\_\{0\}^\{\(i\)\}\-a\_\{i\}x\_\{0\}\\right\|^\{2\}\.\(84\)
Using the stationary moving\-average representation

xs=∑m=0∞Gm​ηs−1−m,x\_\{s\}=\\sum\_\{m=0\}^\{\\infty\}G\_\{m\}\\eta\_\{s\-1\-m\},we have

g0\(i\)=∑j=1∞di,j​x1−j\(i\)=∑ℓ≥1hi,ℓ⊤​η−ℓ,g\_\{0\}^\{\(i\)\}=\\sum\_\{j=1\}^\{\\infty\}d\_\{i,j\}x\_\{1\-j\}^\{\(i\)\}=\\sum\_\{\\ell\\geq 1\}h\_\{i,\\ell\}^\{\\top\}\\eta\_\{\-\\ell\},where

hi,ℓ⊤:=∑j=1ℓdi,j​ei⊤​Gℓ−j\.h\_\{i,\\ell\}^\{\\top\}:=\\sum\_\{j=1\}^\{\\ell\}d\_\{i,j\}\\,e\_\{i\}^\{\\top\}G\_\{\\ell\-j\}\.Similarly,

ai​x0=∑ℓ=1∞ai​Gℓ−1​η−ℓ\.a\_\{i\}x\_\{0\}=\\sum\_\{\\ell=1\}^\{\\infty\}a\_\{i\}G\_\{\\ell\-1\}\\eta\_\{\-\\ell\}\.Therefore,

g0\(i\)−ai​x0=∑ℓ=1∞\(hi,ℓ⊤−ai​Gℓ−1\)​η−ℓ\.g\_\{0\}^\{\(i\)\}\-a\_\{i\}x\_\{0\}=\\sum\_\{\\ell=1\}^\{\\infty\}\\bigl\(h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\bigr\)\\eta\_\{\-\\ell\}\.Since the innovations are independent with covarianceσ2​In\\sigma^\{2\}I\_\{n\},

𝔼​\|g0\(i\)−ai​x0\|2=σ2​∑ℓ=1∞‖hi,ℓ⊤−ai​Gℓ−1‖22\.\\mathbb\{E\}\\left\|g\_\{0\}^\{\(i\)\}\-a\_\{i\}x\_\{0\}\\right\|^\{2\}=\\sigma^\{2\}\\sum\_\{\\ell=1\}^\{\\infty\}\\left\\\|h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\right\\\|\_\{2\}^\{2\}\.\(85\)
Next define

Di​\(z\):=∑j=1∞di,j​zj\.D\_\{i\}\(z\):=\\sum\_\{j=1\}^\{\\infty\}d\_\{i,j\}z^\{j\}\.Since

∑j=0∞ψ⁡\(α,j\)​zj=\(1−z\)α,\\sum\_\{j=0\}^\{\\infty\}\\psi\(\\alpha,j\)z^\{j\}=\(1\-z\)^\{\\alpha\},we obtain

Di​\(z\)=∂α\(1−z\)α\|α=αi,⋆=\(1−z\)αi,⋆​log⁡\(1−z\)\.D\_\{i\}\(z\)=\\left\.\\partial\_\{\\alpha\}\(1\-z\)^\{\\alpha\}\\right\|\_\{\\alpha=\\alpha\_\{i,\\star\}\}=\(1\-z\)^\{\\alpha\_\{i,\\star\}\}\\log\(1\-z\)\.Moreover,

∑ℓ≥1hi,ℓ⊤​zℓ=Di​\(z\)​ei⊤​G​\(z\),\\sum\_\{\\ell\\geq 1\}h\_\{i,\\ell\}^\{\\top\}z^\{\\ell\}=D\_\{i\}\(z\)e\_\{i\}^\{\\top\}G\(z\),whereas

∑ℓ≥1ai​Gℓ−1​zℓ=ai​z​G​\(z\)\.\\sum\_\{\\ell\\geq 1\}a\_\{i\}G\_\{\\ell\-1\}z^\{\\ell\}=a\_\{i\}zG\(z\)\.Thus,

∑ℓ≥1\(hi,ℓ⊤−ai​Gℓ−1\)​zℓ=\(Di​\(z\)​ei⊤−ai​z\)​G​\(z\)\.\\sum\_\{\\ell\\geq 1\}\\bigl\(h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\bigr\)z^\{\\ell\}=\\bigl\(D\_\{i\}\(z\)e\_\{i\}^\{\\top\}\-a\_\{i\}z\\bigr\)G\(z\)\.
By Parseval’s identity,

∑ℓ≥1‖hi,ℓ⊤−ai​Gℓ−1‖22=12​π​∫02​π‖\(Di​\(ei​θ\)​ei⊤−ai​ei​θ\)​G​\(ei​θ\)‖22​𝑑θ\.\\displaystyle\\sum\_\{\\ell\\geq 1\}\\left\\\|h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\right\\\|\_\{2\}^\{2\}=\\frac\{1\}\{2\\pi\}\\int\_\{0\}^\{2\\pi\}\\left\\\|\\bigl\(D\_\{i\}\(e^\{\\mathrm\{i\}\\theta\}\)e\_\{i\}^\{\\top\}\-a\_\{i\}e^\{\\mathrm\{i\}\\theta\}\\bigr\)G\(e^\{\\mathrm\{i\}\\theta\}\)\\right\\\|\_\{2\}^\{2\}\\,d\\theta\.\(86\)
By[Assumption1](https://arxiv.org/html/2609.18127#Thmassumption1),G⁡\(z\)G\(z\)is invertible for every\|z\|=1\|z\|=1\. SinceGGis continuous on the unit circle,

sG=inf\|z\|=1σmin​\(G⁡\(z\)\)\>0\.s\_\{G\}=\\inf\_\{\|z\|=1\}\\sigma\_\{\\min\}\(G\(z\)\)\>0\.Hence, using

‖M​N‖2≥σmin​\(N\)​‖M‖2,\\\|MN\\\|\_\{2\}\\geq\\sigma\_\{\\min\}\(N\)\\\|M\\\|\_\{2\},\([86](https://arxiv.org/html/2609.18127#A4.E86)\) gives

∑ℓ≥1‖hi,ℓ⊤−ai​Gℓ−1‖22≥sG2​12​π​∫02​π‖Di​\(ei​θ\)​ei⊤−ai​ei​θ‖22​𝑑θ\.\\displaystyle\\sum\_\{\\ell\\geq 1\}\\left\\\|h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\right\\\|\_\{2\}^\{2\}\\geq s\_\{G\}^\{2\}\\frac\{1\}\{2\\pi\}\\int\_\{0\}^\{2\\pi\}\\left\\\|D\_\{i\}\(e^\{\\mathrm\{i\}\\theta\}\)e\_\{i\}^\{\\top\}\-a\_\{i\}e^\{\\mathrm\{i\}\\theta\}\\right\\\|\_\{2\}^\{2\}\\,d\\theta\.
Applying Parseval’s identity once more,

∑ℓ≥1‖hi,ℓ⊤−ai​Gℓ−1‖22≥sG2​\[‖di,1​ei⊤−ai‖22\+∑j=2∞di,j2\]\.\\displaystyle\\sum\_\{\\ell\\geq 1\}\\left\\\|h\_\{i,\\ell\}^\{\\top\}\-a\_\{i\}G\_\{\\ell\-1\}\\right\\\|\_\{2\}^\{2\}\\geq s\_\{G\}^\{2\}\\left\[\\left\\\|d\_\{i,1\}e\_\{i\}^\{\\top\}\-a\_\{i\}\\right\\\|\_\{2\}^\{2\}\+\\sum\_\{j=2\}^\{\\infty\}d\_\{i,j\}^\{2\}\\right\]\.\(87\)Combining \([84](https://arxiv.org/html/2609.18127#A4.E84)\), \([85](https://arxiv.org/html/2609.18127#A4.E85)\), and \([87](https://arxiv.org/html/2609.18127#A4.E87)\), and minimizing overai∈ℝ1×na\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}, yields

μt,i\\displaystyle\\mu\_\{t,i\}≥σ2​sG2​infai∈ℝ1×n\[‖di,1​ei⊤−ai‖22\+∑j=2∞di,j2\]\\displaystyle\\geq\\sigma^\{2\}s\_\{G\}^\{2\}\\inf\_\{a\_\{i\}\\in\\mathbb\{R\}^\{1\\times n\}\}\\left\[\\left\\\|d\_\{i,1\}e\_\{i\}^\{\\top\}\-a\_\{i\}\\right\\\|\_\{2\}^\{2\}\+\\sum\_\{j=2\}^\{\\infty\}d\_\{i,j\}^\{2\}\\right\]=σ2​sG2​∑j=2∞di,j2\.\\displaystyle=\\sigma^\{2\}s\_\{G\}^\{2\}\\sum\_\{j=2\}^\{\\infty\}d\_\{i,j\}^\{2\}\.\(88\)
For the first two terms in this sum,

ψ⁡\(α,2\)=α⁡\(α−1\)2,ψ⁡\(α,3\)=−α​\(α−1\)​\(α−2\)6\.\\psi\(\\alpha,2\)=\\frac\{\\alpha\(\\alpha\-1\)\}\{2\},\\qquad\\psi\(\\alpha,3\)=\-\\frac\{\\alpha\(\\alpha\-1\)\(\\alpha\-2\)\}\{6\}\.Therefore,

di,2=αi,⋆−12,d\_\{i,2\}=\\alpha\_\{i,\\star\}\-\\frac\{1\}\{2\},and

di,3=−3​αi,⋆2\+6​αi,⋆−26\.d\_\{i,3\}=\\frac\{\-3\\alpha\_\{i,\\star\}^\{2\}\+6\\alpha\_\{i,\\star\}\-2\}\{6\}\.It follows from \([88](https://arxiv.org/html/2609.18127#A4.E88)\) that

μt,i≥σ2​sG2​\[\(αi,⋆−12\)2\+\(−3​αi,⋆2\+6​αi,⋆−26\)2\]\.\\displaystyle\\mu\_\{t,i\}\\geq\\sigma^\{2\}s\_\{G\}^\{2\}\\left\[\\left\(\\alpha\_\{i,\\star\}\-\\frac\{1\}\{2\}\\right\)^\{2\}\+\\left\(\\frac\{\-3\\alpha\_\{i,\\star\}^\{2\}\+6\\alpha\_\{i,\\star\}\-2\}\{6\}\\right\)^\{2\}\\right\]\.\(89\)
It remains to lower boundsGs\_\{G\}\. For\|z\|=1\|z\|=1,

G​\(z\)−1=diag⁡\(\(1−z\)α1,⋆,…,\(1−z\)αn,⋆\)−z​A⋆\.G\(z\)^\{\-1\}=\\operatorname\{diag\}\\left\(\(1\-z\)^\{\\alpha\_\{1,\\star\}\},\\ldots,\(1\-z\)^\{\\alpha\_\{n,\\star\}\}\\right\)\-zA\_\{\\star\}\.Hence

‖G​\(z\)−1‖op\\displaystyle\\\|G\(z\)^\{\-1\}\\\|\_\{\\mathrm\{op\}\}≤maxi∈\[n\]⁡\|1−z\|αi,⋆\+‖A⋆‖op\\displaystyle\\leq\\max\_\{i\\in\[n\]\}\|1\-z\|^\{\\alpha\_\{i,\\star\}\}\+\\\|A\_\{\\star\}\\\|\_\{\\mathrm\{op\}\}≤2αmax\+‖A⋆‖op,\\displaystyle\\leq 2^\{\\alpha\_\{\\max\}\}\+\\\|A\_\{\\star\}\\\|\_\{\\mathrm\{op\}\},because\|1−z\|≤2\|1\-z\|\\leq 2on the unit circle\. Therefore,

sG=inf\|z\|=11‖G​\(z\)−1‖op≥12αmax\+‖A⋆‖op\.s\_\{G\}=\\inf\_\{\|z\|=1\}\\frac\{1\}\{\\\|G\(z\)^\{\-1\}\\\|\_\{\\mathrm\{op\}\}\}\\geq\\frac\{1\}\{2^\{\\alpha\_\{\\max\}\}\+\\\|A\_\{\\star\}\\\|\_\{\\mathrm\{op\}\}\}\.Substituting this bound into \([89](https://arxiv.org/html/2609.18127#A4.E89)\) completes the proof\. ∎