Conditioned limit theorems for products of random matrices

Abstract

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ behaves like a sum of i.i.d.\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logarithm and the central limit theorem. Denote by $\mathbb{B}$ the closed unit ball in $\mathbb{R}^{d}$ and by $\mathbb{B}^{c}$ its complement. For any $v\in \mathbb{B}^{c}$ define the exit time of the random process $G_{n}v$ from $\mathbb{B}^{c}$ by $\tau_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{\tau_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $\tau_{v}>n.$