Berry–Esseen type bounds for the left random walk on GLd(R) under polynomial moment conditions

Abstract

with common distribution µ. In this paper, under standard assumptions on µ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for log( A n ) when µ has a polynomial moment. More precisely, we get the rate ((log n)/n) q/2−1 when µ has a moment of order q ∈]2, 3] and the rate 1/ √ n when µ has a moment of order 4, which significantly improves earlier results in this setting.

Limit theorems for iid products of positive matrices

“…Taking m ∼ C log n, with C| log(1 − γ)| > 1/2, we infer that the right-hand side is bounded by D/ √ n, and we conclude thanks to Lemma 2.1 of [16], using Theorem 7.1.…”

“…Proof. Applying Proposition 3.2 (with p = q) and Theorem 7.1, we see that we can use Lemma 2.1 of [16] with…”

“…Those coefficients have been introduced in [10], in the setting of products of iid matrices in GL d (R), and proved to be very useful in [13] and [16], see also [12].…”

“…In a series of paper [10], [12], [13], [16] and [17] we studied the stochastic properties of the norm cocycle associated with the left random walk on GL d (R) under optimal or close to optimal moment conditions. The moment conditions are optimal in case of the central limit theorem (CLT) and the ASIP with rate and close to optimal in the case of the Berry-Esseen theorem.…”

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

“…Unlike Theorem 4.2, it is not simple to deduce the Berry–Esseen type bound for $$\log \Vert M_n\Vert$$ from that of $$\log \Vert M_n v\Vert$$, the latter was proven by Bougerol [11]. Indeed, even in the independent and identically distributed case, although the Berry–Esseen bound for $$\log \Vert M_n v\Vert$$ has been known since the work of Le Page [50], the bounds for the matrix norm were only recently studied [28, 29, 67]. Below, we give a version of these results for the Markovian case adapting the approach of Xiao–Grama–Liu [67] and using our large deviation estimates (replacing the large deviation ingredient of [67] from [8] in the independent and identically distributed case).…”