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Sharp Error Terms for Return Time Statistics under Mixing Conditions

Author

Listed:
  • Miguel Abadi

    (Universidade Estadual de Campinas)

  • Nicolas Vergne

    (UR341 Mathématiques et Informatique Appliquées)

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.

Suggested Citation

  • Miguel Abadi & Nicolas Vergne, 2009. "Sharp Error Terms for Return Time Statistics under Mixing Conditions," Journal of Theoretical Probability, Springer, vol. 22(1), pages 18-37, March.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:1:d:10.1007_s10959-008-0199-x
    DOI: 10.1007/s10959-008-0199-x
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