Hitting and returning to rare events for all alpha-mixing processes |
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Authors: | Miguel Abadi |
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Affiliation: | a Universidade de São Paulo, São Paulo, Brazilb Laboratoire de Mathématiques CNRS UMR 6205, Université de Bretagne Occidentale, Brest, France |
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Abstract: | We prove that for any α-mixing stationary process the hitting time of any n-string An converges, when suitably normalized, to an exponential law. We identify the normalization constant λ(An). 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 of 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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Keywords: | Mixing processes Hitting times Repetition times Return times Rare event Exponential approximation |
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