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

Hui Xiao; Ion Grama; Quansheng Liu

paper

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

arXiv:2209.03158

Abstract

Consider the matrix products $G_{n} := g_{n} \dots g_{1}$ , 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 $ρ (G_{n})$ 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.

46 pages, to appear in Journal of Theoretical Probability