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 $v\in {\mathbb{R}}^d\setminus \{0\}.$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert {)}_{n\ge 1}$ 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 ${\tau }_v=\min \{n\ge 1:G_{n}v\in \mathbb{B}\}.$ 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 ${\tau }_v>n.$

References in corpus (3)

• Random walks in cones
• On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains
• Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions