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Exact distribution of word occurrences in a random sequence of letters

Part of: Markov processes Combinatorial probability Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

S. Robin  and
J. J. Daudin
S. Robin*
Affiliation:
INA-PG, Paris
J. J. Daudin*
Affiliation:
INA-PG, Paris
*
∗Postal address: INA-PG – INRA, 16, rue Claude Bernard, 75005 Paris, France.
∗Postal address: INA-PG – INRA, 16, rue Claude Bernard, 75005 Paris, France.
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Abstract

The study of the distribution of the distance between words in a random sequence of letters is interesting in view of application in genome sequence analysis. In this paper we give the exact distribution probability and cumulative distribution function of the distances between two successive occurrences of a given word and between the nth and the (n+m)th occurrences under three models of generation of the letters: i.i.d. with the same probability for each letter, i.i.d. with different probabilities and Markov process. The generating function and the first two moments are also given. The point of studying the distances instead of the counting process is that we get some knowledge not only about the frequency of a word but also about its longitudinal distribution in the sequence.

Keywords

Distance between occurrencesgenome sequence analysisMarkov chainpatternswaiting time

MSC classification

Primary: 60C05: Combinatorial probability 60E05: Distributions
Secondary: 60G50: Sums of independent random variables; random walks 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 179 - 193
Copyright
Copyright © Applied Probability Trust 1999 

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