Conditioned limit theorems for products of positive random matrices

Abstract

Inspired by a recent paper of I. Grama, E. Le Page and M. Peigné, we consider a sequence $(g_n{)}_{n\ge 1}$ of i.i.d. random $d\times d$-matrices with non-negative entries and study the fluctuations of the process $(\log |g_n\cdots g_1\cdot x|{)}_{n\ge 1}$ 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 $\frac{c}{\sqrt{n}}$ for some positive constant $c$.

I would like to replace this article by the latest version in which I: 1. added references. 2. corrected misprints and omitted un neccessary numbers. 3. adjusted the hypothese P2 for positive matrices. 4. added section 5 for the verification of conditions C1-C3 used in section 4. Thank you a lot

References in corpus (2)

• Random walks in cones
• On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains