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hal-00462595, version 1

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

Miguel Abadi () 1, Benoit Saussol () 2

Résumé : 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
  • Versions disponibles :  v1 (25-03-2010) v2 (31-05-2010)
 
  • hal-00462595, version 1
  • http://hal.archives-ouvertes.fr/hal-00462595
  • oai:hal.archives-ouvertes.fr:hal-00462595
  • Contributeur : Benoit Saussol <>
  • Soumis le : Mercredi 10 Mars 2010, 12:00:42
  • Dernière modification le : Jeudi 25 Mars 2010, 12:05:17