CONDITIONAL LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES

Abstract : Consider 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.'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 $\tau _{v}=\min \left\{n\geq 1:G_{n}v\in \mathbb{B}\right\} .$ 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 $\tau _{v}>n.$
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https://hal.archives-ouvertes.fr/hal-01264996
Contributor : Ion Grama <>
Submitted on : Saturday, January 30, 2016 - 9:15:41 AM
Last modification on : Friday, February 19, 2021 - 4:10:03 PM

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• HAL Id : hal-01264996, version 1

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Ion Grama, Émile Le Page, Marc Peigné. CONDITIONAL LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES. Probability Theory and Related Fields, Springer Verlag, 2017, 168, pp.601-639. ⟨hal-01264996⟩

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