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Rates of convergence in invariance principles for random walks on linear groups via martingale methods
Probability · Submitted 18 Sep 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups.
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In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of Wasserstein distance and in the almost sure invariance principle for R d-valued martingales with stationary increments having moments of order p $\in$]2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. | contribution 0.90 | 0.95 | 0.75 | asserted 1.00 |
| s1 | Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. | method 1.00 | 0.87 | 0.04 | asserted 1.00 |
| s2 | Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation. | gap 0.99 | 0.07 | 0.04 | asserted 1.00 |
| s3 | A part of this paper is devoted to circumvent this issue. | contribution 0.79 | 0.50 | 0.03 | asserted 0.88 |
| s4 | We then exhibit near optimal rates both in the central limit theorem in terms of Wasserstein distance and in the almost sure invariance principle for R d-valued martingales with stationary increments having moments of order p $\in$]2, 3] (the case of sequences of reversed martingale differences is also considered). | result 0.91 | 0.96 | 0.52 | asserted 1.00 |
| s5 | Note also that, as an application of our results for general R d-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups. | contribution 0.92 | 0.86 | 0.05 | asserted 0.98 |
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Original abstract
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of Wasserstein distance and in the almost sure invariance principle for R d-valued martingales with stationary increments having moments of order p $\in$]2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups.
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We also mention that our approach opens a way to study invariance principles for the coefficients $⟨f, G_n v ⟩$; recent progress in the study of invariance principles can be found in Cuny, Dedecker and Jan [ref] and Cuny, Dedecker and Merlevède [ref] , where the vector norm $|G_n v|$ and the operator norm $G_n$ have been studied via the martingale approximation approach.
Large deviation expansions for the coefficients of random walks on the general linear group · Brief overview of the main results · 0.85
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