Edgeworth expansion and large deviations for the coefficients of products of positive random matrices

TL;DR

This paper develops precise probabilistic approximations and large deviation results for entries and spectral radius of products of positive random matrices, under optimal moment conditions, using spectral gap theory and Edgeworth expansions.

Contribution

It introduces new Edgeworth expansions and large deviation asymptotics for matrix entries and spectral radius, under minimal moment assumptions, with applications to local limit theorems.

Findings

Established Berry-Esseen theorem and Edgeworth expansion for matrix entries

Derived large deviation asymptotics for entries and spectral radius

Developed spectral gap theory for the norm cocycle and coefficients

Abstract

Consider the matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices. Under the optimal third moment condition, we first establish a Berry-Esseen theorem and an Edgeworth expansion for the $(i,j)$-th entry $G_n^{i,j}$ of the matrix $G_n$, where $1 \leq i, j \leq d$. Using the Edgeworth expansion for $G_n^{i,j}$ under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries $G_n^{i,j}$ subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for $G_n^{i,j}$ and upper and lower large deviations bounds for the spectral radius $\rho(G_n)$ of $G_n$. A byproduct of our approach is the local limit theorem for $G_n^{i,j}$ under the optimal second moment condition. In the proofs we develop a spectral gap theory for the norm cocycle and for the coefficients, which is of independent interest.