Counts of Failure Strings in Certain Bernoulli Sequences

Abstract

In a sequence of independent Bernoulli trials the probability for success in thekth trial ispk,k= 1, 2, …. 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 whichpk=a/(a+b+k− 1),a> 0,b≥ 0. Also, the limit behaviour of the longest failure string in the firstntrials is considered. Forb= 0, the strings correspond to cycles in random permutations.