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Journal of Theoretical Probability

, Volume 22, Issue 1, pp 18–37 | Cite as

Sharp Error Terms for Return Time Statistics under Mixing Conditions

  • Miguel AbadiEmail author
  • Nicolas Vergne
Article

Abstract

We describe the statistics of repetition times of a string of symbols in a stochastic process.

Denote by τ A the time elapsed until the process spells a finite string A and by S A the number of consecutive repetitions of A. We prove that, if the length of the string grows unboundedly, (1) the distribution of τ A , when the process starts with A, is well approximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S A is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and also allow us to get approximations for all the moments of τ A and S A . To obtain (1) we assume that the process is φ-mixing, while to obtain (2) we assume the convergence of certain conditional probabilities.

Keywords

Mixing Recurrence Rare event Return time Sojourn time 

Mathematics Subject Classification (2000)

60F05 60G10 37A50 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.IMECCUniversidade Estadual de CampinasCampinasBrazil
  2. 2.INRAUR341 Mathématiques et Informatique AppliquéesJouy-en-Josas CedexFrance

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