Sharp error terms for return time statistics under mixing conditions *

Miguel Abadi (1) , Nicolas Vergne (2, 3)

1 IME - Instituto de Matemática e Estatística
2 MIA - Unité de recherche Mathématiques et Informatique Appliquées
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

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.

Mots clés: Mixing; recurrence; rare event; return time; sojourn time

Domaines: Applications [stat.AP]; Méthodologie [stat.ME]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Miguel Abadi, Nicolas Vergne. Sharp error terms for return time statistics under mixing conditions *. Journal of Theoretical Probability, 2009. ⟨hal-02337198⟩