Conditioned limit theorems for products of random matrices

Consider the product $G_n=g_n\mathrm{.}\mathrm{.}\mathrm{.}{g}_1$ of the random matrices $g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_n$ in $GL(d,\mathbb{R})$ and the random process $G_{n}v=g_n\mathrm{.}\mathrm{.}\mathrm{.}{g}_{1}v$ in ${\mathbb{R}}^d$ starting at point $v\in $\mathbb{R}$^{d}\smallsetminus {0} .$ It is well known that under appropriate assumptions, the sequence $(\log \mathrm{\parallel }{G}_{n}v\mathrm{\parallel }{)}_{n\ge 1}$ 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\geq 1:G_{n}v\in $\mathbb{B}$} .$Weestablishtheasymptoticas$n \to \infty $oftheprobabilityoftheevent${\tau_{v}>n} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}}\log \mathrm{\parallel }{G}_{n}v\mathrm{\parallel }$ conditioned that ${\tau }_v>n\mathrm{.}$