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Conditioned limit theorems for products of random matrices. (English) Zbl 1406.60015
Summary: Let \(g_{1},g_{2},\ldots \) be i.i.d. random matrices in \(\mathrm{GL}(d,\mathbb {R})\). For any \(n\geq 1\) consider the product \(G_{n}=g_{n} \dotsc g_{1}\) and the random process \(G_{n}v=g_{n}\dotsc g_{1}v\) in \(\mathbb {R}^{d}\) starting at point \(v\in \mathbb {R}^{d}\setminus \{0\} .\) It is well known that under appropriate assumptions, the sequence \(\left(\log \| G_{n}v\|\right) _{n\geq 1}\) behaves like a sum of i.i.d. r.v.’s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with \(\| v \| >1\) 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 \(\frac{1}{\sqrt{n}} \log \| G_{n}v\| \) conditioned that \(\tau _{v}>n\).

MSC:
60B20 Random matrices (probabilistic aspects)
60J05 Discrete-time Markov processes on general state spaces
60J45 Probabilistic potential theory
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