Joint Distributions of Counts of Strings in Finite Bernoulli Sequences
| Autor: | Fred W. Huffer, Jayaram Sethuraman |
|---|---|
| Rok vydání: | 2012 |
| Předmět: |
Discrete mathematics
Statistics and Probability Sequence Factorial counts of strings flaws and failures Multivariate random variable General Mathematics random permutation 010102 general mathematics cycles Bernoulli sequence Random permutation 01 natural sciences Combinatorics Bernoulli's principle Joint probability distribution Conditional marked Poisson process 60K99 60C05 Bernoulli process 0101 mathematics Statistics Probability and Uncertainty Random variable Mathematics |
| Zdroj: | J. Appl. Probab. 49, no. 3 (2012), 758-772 |
| ISSN: | 1475-6072 0021-9002 |
| DOI: | 10.1017/s0021900200009529 |
| Popis: | An infinite sequence (Y 1, Y 2,…) of independent Bernoulli random variables with P(Y i = 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 Z 1, Z 2, Z 3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z 1, Z 2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1, Y 2,…, Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1, 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. |
| Databáze: | OpenAIRE |
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