Limit theorems for iid products of positive matrices

TL;DR

This paper establishes limit theorems such as the ASIP and Berry-Esseen for iid products of positive matrices, revealing optimal rates under specific moment conditions and demonstrating exponential decay of coupling coefficients.

Contribution

It provides the first optimal rate results for the ASIP and Berry-Esseen for iid positive matrix products, including cases with invariant cones.

Findings

ASIP with rate o(n^{1/p}) for p > 2

Berry-Esseen theorem with rate O(1/√n) for third moment

Exponential decay of coupling coefficients in positive matrices

Abstract

We study stochastic properties of the norm cocycle associated with iid products of positive matrices. We obtain the almost sure invariance principle (ASIP) with rate o(n 1/p) under the optimal condition of a moment or order p > 2 and the Berry-Esseen theorem with rate O(1/ $\sqrt$ n) under the optimal condition of a moment of order 3. The results are also valid for the matrix norm. For the matrix coefficients, we also have the ASIP but we obtain only partial results for the Berry-Esseen theorem. The proofs make use of coupling coefficients that surprisingly decay exponentially fast to 0 while there is only a polynomial decay in the case of invertible matrices. All the results are actually valid in the context of iid products of matrices leaving invariant a suitable cone.