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- W2266948488 abstract "Let $$g_{1},g_{2},dots $$ be i.i.d. random matrices in $$GLleft( d,mathbb {R}right) .$$ For any $$nge 1$$ consider the product $$G_{n}=g_{n} dots g_{1}$$ and the random process $$G_{n}v=g_{n}dots g_{1}v$$ in $$mathbb {R}^{d}$$ starting at point $$vin mathbb {R}^{d}{backslash } left{ 0right} .$$ It is well known that under appropriate assumptions, the sequence $$left( log left| G_{n}vright| right) _{nge 1}$$ 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 $$left| v right| >1$$ 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 $$nrightarrow +infty $$ of the probability of the event $$left{ tau _{v}>nright} $$ and find the limit law for the quantity $$frac{1}{sqrt{n}} log left| G_{n}vright| $$ conditioned that $$tau _{v}>n$$ ." @default.
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- W2266948488 date "2016-05-31" @default.
- W2266948488 modified "2023-09-25" @default.
- W2266948488 title "Conditioned limit theorems for products of random matrices" @default.
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- W2266948488 doi "https://doi.org/10.1007/s00440-016-0719-z" @default.
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