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Exact distribution of word occurrences in a random sequence of letters

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Stéphane ROBIN and Jean-Jacques DAUDIN

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*Journal of Applied Probability*, vol. 36, 179-193, 1999.

**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 *n*-th and the (*n*+*m*)-th
occurences under three models of generation of the letters : iid with the
same probability for each letter, iid with different probabilities and
Markov process. The generating function and the first two moments are also
given. The point of studying the distances in place 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.

**Key words and phrases**
distance between occurrences, genome sequence analysis,
Markov chain, patterns, waiting time.