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Joint distributions of counts of strings in finite Bernoulli sequences. (English) Zbl 1314.60034

Summary: An infinite sequence \((Y_{1}, Y_{2},\dots )\) of independent Bernoulli random variables with \(P(Y_{i} = 1) = a / (a + b + i - 1)\), \(i = 1, 2,\dots \), where \(a > 0\) and \(b \geq 0\), will be called a \(\mathrm{Bern}(a, b)\) sequence. Consider the counts \(Z_{1}, Z_{2}, Z_{3},\dots \) of occurrences of patterns or strings of the form \(\{11\}, \{101\}, \{1001\},\dots\), respectively, in this sequence. The joint distribution of the counts \(Z_{1}, Z_{2},\dots \) in the infinite \(\mathrm{Bern}(a, b)\) sequence has been studied extensively. The counts from the initial finite sequence \((Y_{1}, Y_{2},\dots , Y_{n})\) have been studied by L. Holst [ibid. 44, No. 3, 824–830 (2007; Zbl 1132.60011); ibid. 45, No. 3, 901–906 (2008; Zbl 1151.60002)], who obtained the joint factorial moments for \(\mathrm{Bern}(a, 0)\) and the factorial moments of \(Z_{1}\), the count of the string \(\{1, 1\}\), for a general \(\mathrm{Bern}(a, b)\). 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 by F. W. Huffer et al. [Proc. Am. Math. Soc. 137, No. 6, 2125–2134 (2009; Zbl 1165.60305)]. Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

MSC:

60C05 Combinatorial probability
60K99 Special processes
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References:

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