Conditioned limit theorems for products of random matrices

Abstract

Let $$g_{1},g_{2},\dots $$ be i.i.d. random matrices in $$GL\left( d,\mathbb {R}\right) .$$ For any $$n\ge 1$$ consider the product $$G_{n}=g_{n} \dots g_{1}$$ and the random process $$G_{n}v=g_{n}\dots g_{1}v$$ in $$\mathbb {R}^{d}$$ starting at point $$v\in \mathbb {R}^{d}{\backslash } \left\{ 0\right\} .$$ It is well known that under appropriate assumptions, the sequence $$\left( \log \left\| G_{n}v\right\| \right) _{n\ge 1}$$ behaves like a sum of i.i.d. r.v.’s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with $$\left\| v \right\| >1$$ denote by $$\tau _v$$ the first time when the random process $$G_{n}v$$ enters the closed unit ball in $$\mathbb {R}^{d}.$$ We establish the asymptotic as $$n\rightarrow +\infty $$ of the probability of the event $$\left\{ \tau _{v}>n\right\} $$ and find the limit law for the quantity $$\frac{1}{\sqrt{n}} \log \left\| G_{n}v\right\| $$ conditioned that $$\tau _{v}>n$$ .