Sharp Error Terms for Return Time Statistics under Mixing Conditions

Miguel Abadi () and Nicolas Vergne ()
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Miguel Abadi: Universidade Estadual de Campinas
Nicolas Vergne: UR341 Mathématiques et Informatique Appliquées

Journal of Theoretical Probability, 2009, vol. 22, issue 1, 18-37

Abstract: 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; 60F05; 60G10; 37A50 (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10959-008-0199-x

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