Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

TL;DR

This paper establishes Berry-Esseen bounds and moderate deviation results for the norm, entries, and spectral radius of products of i.i.d. positive random matrices, advancing understanding of their probabilistic behavior.

Contribution

It provides the first Berry-Esseen bounds and moderate deviation expansions for these matrix functionals under suitable conditions.

Findings

Berry-Esseen bounds for matrix norm and spectral radius

Moderate deviation expansions of Cramér type for entries and spectral radius

Quantitative convergence rates in the central limit theorem for matrix products

Abstract

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and moderate deviation expansions of Cram\'er type, for the matrix norm $\| G_n \|$ of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectral radius $\rho(G_n)$.