# Precise large deviation asymptotics for products of random matrices

## TL;DR

This paper derives precise large deviation asymptotics for the norms of products of i.i.d. random matrices, extending existing results and providing new limit theorems for matrix norms and related quantities.

## Contribution

It establishes exact asymptotics for large deviation probabilities of matrix products, including invertible and positive matrices, and improves prior large deviation principles.

## Key findings

- Precise asymptotics for large deviations of $\log | G_n x |\
- Enhanced large deviation principles for matrix norms
- Derived a local limit theorem with large deviations

## Abstract

Let $(g_{n})_{n\geq 1}$ be a sequence of independent identically distributed $d\times d$ real random matrices with Lyapunov exponent $\gamma$. For any starting point $x$ on the unit sphere in $\mathbb R^d$, we deal with the norm $ | G_n x | $, where $G_{n}:=g_{n} \ldots g_{1}$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb P(\log | G_n x | \geq n(q+l))$, where $q>\gamma $ is fixed and $l$ is vanishing as $n\to \infty$. We study both invertible matrices and positive matrices and give analogous results for the couple $(X_n^x,\log | G_n x |)$ with target functions, where $X_n^x= G_n x /| G_n x |$. As applications we improve previous results on the large deviation principle for the matrix norm $\|G_n\|$ and obtain a precise local limit theorem with large deviations.