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Sharp error terms for return time statistics under mixing conditions *

Abstract : We describe the statistics of repetition times of a string of symbols in a stochastic process. We consider a string A of length n and prove: 1) The time elapsed until the process starting with A repeats A, denoted by τA, has a distribution which can be well approximated by a degenerated law at the origin and an exponential law. 2) The number of consecutive repetitions of A, denoted by SA, has a distribution which is approximately a geometric law. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of τA and SA. Our results hold for processes that verify the φ-mixing condition.
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Miguel Abadi, Nicolas Vergne. Sharp error terms for return time statistics under mixing conditions *. Journal of Theoretical Probability, Springer, 2009. ⟨hal-02337198⟩

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