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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
##### Keywords:
exit time; Markov chains; random matrices; spectral gap
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