Limit theorems for the coefficients of random walks on the general linear group

2021/11/20 by Hui Xiao, Ion Grama, Xiao, Hui +3
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2111.10569

openalex publication_date 2021/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (g_n)_{n≥ 1} be a sequence of independent and identically distributed random elements with law μ on the general linear group \textrmGL(V), where V=\mathbb R^d. Consider the random walk G_n : = g_n … g₁, n ≥ 1, and the coefficients ⟨ f, G_n v ⟩, where v ∈ V and f ∈ V^*. Under suitable moment assumptions on μ, we prove the strong and weak laws of large numbers and the central limit theorem for ⟨ f, G_n v ⟩, which improve the previous results established under the exponential moment condition on μ. We further demonstrate the Berry-Esseen bound, the Edgeworth expansion, the Cramér type moderate deviation expansion and the local limit theorem with moderate deviations for ⟨ f, G_n v ⟩ under the exponential moment condition. Under a subexponential moment condition on μ, we also show a Berry-Esseen type bound and the moderate deviation principle for ⟨ f, G_n v ⟩. Our approach is based on various versions of the Hölder regularity of the invariant measure of the Markov chain G_n ⋅ x = \mathbb R G_n v on the projective space of V with the starting point x = \mathbb R v.

Citations

• Products of Random Matrices
• Random difference equations and Renewal theory for products of random matrices
• Parametric Furstenberg Theorem on random products of S L ( 2 , R ) matrices
• Berry-Esseen bound and Local Limit Theorem for the coefficients of products of random matrices
• Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GLd(R)

Related

• Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group
• Large deviation expansions for the coefficients of random walks on the general linear group
• Edgeworth expansion for the coefficients of random walks on the general linear group
• Large and moderate deviations for the left random walk on GL d (R)
• Limit theorems for inhomogeneous random walks on GL(d,\mathbb R)