Limit Theorems for the Left Random Walk on GLd (R)

TL;DR

This paper establishes limit theorems like the strong law, CLT, and invariance principles for products of i.i.d. matrices in GLd(R), extending previous work and providing rates of convergence.

Contribution

It extends existing results by proving new limit theorems with rates for the left random walk on GLd(R), building on recent research and prior PhD thesis work.

Findings

Proved Marcinkiewicz-Zygmund strong law for GLd(R)

Established CLT with convergence rates in Wasserstein distances

Demonstrated almost sure invariance principles with explicit rates

Abstract

Motivated by a recent work of Benoist and Quint and extending results from the PhD thesis of the third author, we obtain limit theorems for products of independent and identically distributed elements of GLd (R), such as the Marcinkiewicz-Zygmund strong law of large numbers, the CLT (with rates in Wasserstein's distances) and almost sure invariance principles with rates.