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On the number of runs for Bernoulli arrays

Part of: Distribution theory - Probability Special processes Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Djilali Ait Aoudia  and
Éric Marchand
Djilali Ait Aoudia*
Affiliation:
Université de Sherbrooke
Éric Marchand*
Affiliation:
Université de Sherbrooke
*
∗Current address: Département de Mathématiques et de Statistique, Université de Montréal, Montréal, QC H3C 3J7, Canada.
∗∗Postal address: Département de Mathématiques, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada. Email address: eric.marchand@usherbrooke.ca
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Abstract

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We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries Xk,j and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of Sn for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

Keywords

BernoullimultinomialPólya urnprobability generating functionruns

MSC classification

Secondary: 60C05: Combinatorial probability 60E05: Distributions 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 367 - 377
Copyright
Copyright © Applied Probability Trust 2010 

References

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