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On Consecutive Records in Certain Bernoulli Sequences

Part of:Special processesCombinatorial 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–100 44 Stockholm, Sweden. Email address: lholst@math.kth.se
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Abstract

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In an infinite sequence of independent Bernoulli trials with success probabilities p k =a/(a+b +k-1) for k=1,2,3,…, let N r be the number of r≥2 consecutive successes. Expressions for the first two moments of N r are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of N r can be calculated for small r.

Keywords

Bernoulli trialembedding in Poisson processrandom permutationrecordsruns of successessums of indicators

MSC classification

Secondary:60C05: Combinatorial probability60K99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009, pp. 1201 - 1208
Copyright
Copyright © Applied Probability Trust 2009 

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