Conditioned limit theorems for products of positive random matrices

TL;DR

This paper investigates the fluctuations of the logarithm of the product of positive random matrices acting on vectors, establishing conditioned limit theorems and decay rates for the probability of staying positive.

Contribution

It introduces a martingale approximation approach and harmonic function analysis to derive conditioned limit theorems for products of positive random matrices.

Findings

Probability to stay positive up to time n decreases as c/√n

Method involves harmonic functions and martingale approximation

Provides conditions for fluctuation behaviors of matrix products

Abstract

Inspired by a recent paper of I. Grama, E. Le Page and M. Peign\'e, we consider a sequence $(g_n)_{n \geq 1}$ of i.i.d. random $d\times d$-matrices with non-negative entries and study the fluctuations of the process $(\log \vert g_n\cdots g_1\cdot x\vert )_{n \geq 1}$ for any non-zero vector $x$ in $\mathbb R^d$ with non-negative coordinates. Our method involves approximating this process by a martingale and studying harmonic functions for its restriction to the upper half line. Under certain conditions, the probability for this process to stay in the upper half real line up to time $n$ decreases as $c \over \sqrt n$ for some positive constant $c$.