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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Abadi, Miguel²  Saussol, Benoit¹ 

[1] Univ Bretagne Occidentale, CNRS, UMR 6205, Math Lab, Brest, France

[2] Univ Sao Paulo, Sao Paulo, Brazil

关键词: Mixing processes;    Hitting times;    Repetition times;    Return times;    Rare event;    Exponential approximation;   

DOI  :  10.1016/j.spa.2010.11.001

来源: Elsevier

【 摘 要 】

We prove that for any a-mixing stationary 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 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. (c) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free