## 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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### References:

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