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

2010/03/25 by Abadi, Miguel, Saussol, Benoit · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1003.4856

Abstract

We prove that for any α-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 λ(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.

Cited by

• Return Times Distribution of Expanding Maps
• Higher order return times for ϕ-mixing measures

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