## 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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### References:

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