Conditioned limit theorems for products of random matrices

TL;DR

This paper studies the asymptotic behavior of products of random matrices acting on vectors, focusing on the probability of remaining outside a unit ball and the distribution of the log-norm conditioned on this event.

Contribution

It establishes the asymptotic probability and limit law for the process conditioned on not entering the unit ball, extending classical results to conditioned scenarios.

Findings

Asymptotic probability of the process remaining outside the unit ball is derived.

Limit law for the scaled log-norm conditioned on survival is obtained.

Results extend classical limit theorems to conditioned random matrix products.

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 $\tau_{v}>n.$