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Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Part of: Combinatorial probability Special processes

Published online by Cambridge University Press:  04 February 2016

Fred W. Huffer  and
Jayaram Sethuraman
Fred W. Huffer*
Affiliation:
Florida State University
Jayaram Sethuraman*
Affiliation:
Florida State University
*
∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA.
∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA.
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Abstract

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An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

Keywords

Conditional marked Poisson processBernoulli sequencecounts of stringsrandom permutationcyclesflaws and failures

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 758 - 772
Copyright
© Applied Probability Trust 

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