On Consecutive Records in Certain Bernoulli Sequences

Abstract

In an infinite sequence of independent Bernoulli trials with success probabilities p k =a/(a+b +k-1) for k=1,2,3,…, let N r be the number of r≥2 consecutive successes. Expressions for the first two moments of N r are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of N r can be calculated for small r.