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On consecutive records in certain Bernoulli sequences. (English) Zbl 1187.60006

Summary: In an infinite sequence of independent Bernoulli trials with success probabilities \(p_k = a/(a+b +k-1)\) for \(k=1,2,3,\cdots \), let \(N_r\) be the number of \(r\geq 2\) 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 \(r\).

MSC:

60C05 Combinatorial probability
60K99 Special processes
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