Hitting and returning into rare events for all alpha-mixing processes

Details Hitting and returning into rare events for all alpha-mixing processes Hitting and returning into rare events for all alpha-mixing processes

2010-03-25

arxiv.org/abs/1003.4856v2

Mathematics

Dynamical Systems

Scientific paper

We prove that for any $\alpha$-mixing stationnary process the hitting time of any $n$-string $A_n$ converges, when suitably normalized, to an exponential law. We identify the normalization constant $\lambda(A_n)$. A similar statement holds also for the return time. To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem by Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any $n$-string in $n$ consecutive observations, goes to zero as $n$ goes to infinity.

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