Sharp Error Terms for Return Time Statistics under Mixing Conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process.Denote by τA the time elapsed until the process spells a finite string A and by SA the number of consecutive repetitions of A. We prove that, if the length of the string grows unboundedly, (1) the distribution of τA, 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) SA 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 τA and SA. To obtain (1) we assume that the process is φ-mixing, while to obtain (2) we assume the convergence of certain conditional probabilities.

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