A study of counts of Bernoulli strings via conditional Poisson processes

2008/01/14 by Huffer, Fred W., Sethuraman, Jayaram, Sethuraman, Sunder
#60C05 #60K99 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0801.2115

Abstract

We say that a string of length d occurs, in a Bernoulli sequence, if a success is followed by exactly (d-1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d-cycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all d-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. This general class includes all Bernoulli sequences considered before, as well many new sequences.

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