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September 2012 Joint distributions of counts of strings in finite Bernoulli sequences
Fred W. Huffer, Jayaram Sethuraman
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J. Appl. Probab. 49(3): 758-772 (September 2012). DOI: 10.1239/jap/1346955332


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.


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Fred W. Huffer. Jayaram Sethuraman. "Joint distributions of counts of strings in finite Bernoulli sequences." J. Appl. Probab. 49 (3) 758 - 772, September 2012.


Published: September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 1314.60034
MathSciNet: MR3012098
Digital Object Identifier: 10.1239/jap/1346955332

Primary: 60C05
Secondary: 60K99

Keywords: Bernoulli sequence , Conditional marked Poisson process , counts of strings , cycles , flaws and failures , random permutation

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 3 • September 2012
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