Counts of failure strings in certain Bernoulli sequences

• In a sequence of independent Bernoulli trials the probability for success in the kth trial is Pk 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 which Pk = a/(a + b + k - 1), a > 0, b >= 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.

Nyckelord

binomial moment

Ewens sampling formula

Hoppe's urn

Poisson

distribution

Poisson-Dirichlet distribution

Polya's urn

random

permutation

record

spacing

sums of indicators

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