Sharp errors for point-wise Poisson approximations in mixing processes

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 the number of occurrences of a string of symbols in a stochastic process: taking a string A of length n, we prove that the number of visits to A up to time t, denoted by N t , has approximately a Poisson distribution. We provide a sharp error for this approximation. In contrast to previous works which present uniform error terms based on the total variation distance, our error is point-wise. As a byproduct we obtain that all the moments of N t are finite. Moreover, we obtain explicit approximations for all of them. Our result holds for processes that verify the φ-mixing condition. The error term is explicitly expressed as a function of the rate function φ and is easily computable.

Mots clés: Poisson distribution; number of visits; Mixing; recurrence; rare event

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

Miguel Abadi, Nicolas Vergne. Sharp errors for point-wise Poisson approximations in mixing processes. Nonlinearity, 2008, 21 (12), pp.2871-2885. ⟨10.1088/0951-7715/21/12/008⟩. ⟨hal-02337199⟩