CONDITIONAL LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES

DOI 10.1007/s00440-016-0719-zInternational audienceConsider the product $G_{n}=g_{n} \dots g_{1}$ of the random matrices $g_{1},\dots,g_{n}$ in $GL\left( d,\mathbb{R}\right) $ and the random process $G_{n}v=g_{n}\dots g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \left\{ 0\right\} .$ It is well known that under appropriate assumptions, the sequence $\left( \log \left\Vert G_{n}v\right\Vert \right) _{n\geq 1}$ behaves like a sum of i.i.d.~r.v.'sand satisfies standard classical properties such as the law of largenumbers, 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 therandom process $G_{n}v$ from $\mathbb{B}^{c}$ by $\tau _{v}=\min \left\{n\geq 1:G_{n}v\in \mathbb{B}\right\} .$ 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\Vert G_{n}v\right\Vert $ conditioned that $\tau _{v}>n.