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

Abstract

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\text{GL}(V)$, where $V={\mathbb{R}}^d$. Consider the random walk $G_n:=g_n\dots g_1$, $n\ge 1$, and the coefficients $⟨f,G_{n}v⟩$, where $v\in V$ and $f\in V^{\ast }$. Under suitable moment assumptions on $\mu$, 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 $\mu$. 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 $\mu$, 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 $G_n\cdot x=\mathbb{R}{G}_{n}v$ on the projective space of $V$ with the starting point $x=\mathbb{R}v$.

66 pages

References in corpus (4)

• Large deviation expansions for the coefficients of random walks on the general linear group
• Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random 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)