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arXiv : 1003.4856

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v1 (25-03-2010)

v2 (31-05-2010)

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

Miguel Abadi 1, Benoit Saussol 2

(10/03/2010)

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.

1 :	Instituto de Matemática e Estatística (IME)
	Universidade de São Paulo
2 :	Laboratoire de mathématiques de Brest (LM)
	CNRS : UMR6205 – Université de Bretagne Occidentale [UBO] – Institut Supérieur des Sciences et Technologies de Brest (ISSTB)

Domaine

:

Mathématiques/Systèmes dynamiques

Mots Clés : alpha mixing – exponential law – return time

hal-00462595, version 2

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Contributeur : Benoit Saussol <>

Soumis le : Jeudi 25 Mars 2010, 16:10:30

Dernière modification le : Lundi 31 Mai 2010, 11:17:45