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

Miguel Abadi (1) , Benoit Saussol (2)

1 IME - Instituto de Matemática e Estatística
2 LM - Laboratoire de mathématiques de Brest

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.

Mots clés: alpha mixing; exponential law; return time

Domaines: Systèmes dynamiques [math.DS]

Miguel Abadi, Benoit Saussol. Hitting and returning into rare events for all alpha-mixing processes. 2010. ⟨hal-00462595v2⟩