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On the First k Moments of the Random Count of a Pattern in a Multistate Sequence Generated by a Markov Source

Part of: Distribution theory Markov processes

Published online by Cambridge University Press:  14 July 2016

G. Nuel
G. Nuel*
Affiliation:
Paris Descartes University
*
∗Postal address: MAP5, Department of Applied Mathematics, CNRS 8145, Paris Descartes University, 49 rue des Saints-Pères, F-75006 Paris, France. Email address: [email protected]
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Abstract

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In this paper we develop an explicit formula that allows us to compute the first k moments of the random count of a pattern in a multistate sequence generated by a Markov source. We derive efficient algorithms that allow us to deal with any pattern (low or high complexity) in any Markov model (homogeneous or not). We then apply these results to the distribution of DNA patterns in genomic sequences, and we show that moment-based developments (namely Edgeworth's expansion and Gram-Charlier type-B series) allow us to improve the reliability of common asymptotic approximations, such as Gaussian or Poisson approximations.

Keywords

Optimal Markov chain embeddingdeterministic finite automatonmoment generating functionEdgeworth's expansionGram-Charlier series

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62E15: Exact distribution theory 62E17: Approximations to distributions (nonasymptotic)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1105 - 1123
Copyright
Copyright © Applied Probability Trust 2010 

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