Large and moderate deviations for the left random walk on GL d (R)

TL;DR

This paper studies the probabilities of large and moderate deviations in the behavior of products of i.i.d. matrices in GL(d,R), providing bounds and principles under various moment conditions.

Contribution

It offers a comprehensive analysis of deviation probabilities for matrix products in GL(d,R) using martingale techniques, covering a range of moment conditions and establishing a moderate deviation principle.

Findings

Derived upper bounds for large deviations

Established moderate deviation principles

Analyzed all moment condition scenarios

Abstract

Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements of GL d (R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of order p \textgreater{} 1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.