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Counts of failure strings in certain Bernoulli sequences. (English) Zbl 1132.60011

Summary: In a sequence of independent Bernoulli trials the probability for success in the \(k\)th trial is \(p_k\), \(k = 1, 2,\dots\). The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in which \(p_k = a/(a + b + k - 1)\), \(a > 0\), \(b\geq 0\). Also, the limit behaviour of the longest failure string in the first \(n\) trials is considered. For \(b = 0\), the strings correspond to cycles in random permutations.

MSC:

60C05 Combinatorial probability
60K99 Special processes
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