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Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

Part of: Distribution theory Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Etienne Roquain  and
Sophie Schbath
Etienne Roquain*
Affiliation:
Institut National de la Recherche Agronomique
Sophie Schbath*
Affiliation:
Institut National de la Recherche Agronomique
*
∗ Postal address: INRA, Unité Mathématique, Informatique et Génome, Domaine de Vilvert, F-78352 Jouy-en-Josas, France.
∗ Postal address: INRA, Unité Mathématique, Informatique et Génome, Domaine de Vilvert, F-78352 Jouy-en-Josas, France.
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Abstract

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We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

Keywords

Compound Poisson approximationChen-Stein methodmultiple word countclumpperiodMarkov chain

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 128 - 140
Copyright
Copyright © Applied Probability Trust 2007 

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