Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

TL;DR

This paper establishes moderate deviation expansions and local limit theorems for the coefficients of random walks on the general linear group, using regularity properties of invariant measures under certain conditions.

Contribution

It introduces new moderate deviation and local limit results for coefficients of random linear group walks, leveraging the regularity of invariant measures.

Findings

Proves Cramér type moderate deviation expansions.

Establishes local limit theorems with moderate deviations.

Uses Hölder regularity of invariant measures for analysis.

Abstract

Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, where $(g_n)_{n\geq 1}$ is a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group ${\rm GL}(V)$ with $V=\mathbb R^d$. Under suitable conditions on $\mu$, we establish Cram\'{e}r type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients $\langle f, G_n v \rangle$, where $v \in V$ and $f \in V^*$. Our approach is based on the H\"older 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 $x = \mathbb R v$, under the changed measure.