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Counts of Failure Strings in Certain Bernoulli Sequences

Part of: Special processes Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Lars Holst
Lars Holst*
Affiliation:
Royal Institute of Technology
*
∗Postal address: Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden. Email address: [email protected]
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Abstract

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In a sequence of independent Bernoulli trials the probability for success in the kth trial is pk, k = 1, 2, …. The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in which pk = a/(a + b + k − 1), a > 0, b ≥ 0. Also, the limit behaviour of the longest failure string in the first n trials is considered. For b = 0, the strings correspond to cycles in random permutations.

Keywords

Binomial momentEwens sampling formulaHoppe's urnPoisson distributionPoisson-Dirichlet distributionPólya's urnrandom permutationrecordspacingsums of indicators

MSC classification

Secondary: 60C05: Combinatorial probability 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 824 - 830
Copyright
Copyright © Applied Probability Trust 2007 

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