Conditioned limit theorems for products of random matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866915071687393280 |
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| author | Grama, Ion Page, Emile Le Peigné, Marc |
| author_facet | Grama, Ion Page, Emile Le Peigné, Marc |
| contents | 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 $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ 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 $τ_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{τ_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $τ_{v}>n.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_0423 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Conditioned limit theorems for products of random matrices Grama, Ion Page, Emile Le Peigné, Marc Probability 60B20, 60J05, 60J45 (Primary), 37A50 (Secondary) 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 $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ 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 $τ_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{τ_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $τ_{v}>n.$ |
| title | Conditioned limit theorems for products of random matrices |
| topic | Probability 60B20, 60J05, 60J45 (Primary), 37A50 (Secondary) |
| url | https://arxiv.org/abs/1411.0423 |