A study of counts of Bernoulli strings via conditional Poisson processes

Huffer, Fred; Sethuraman, Jayaram; Sethuraman, Sunder

doi:10.1090/S0002-9939-08-09793-1

A study of counts of Bernoulli strings via conditional Poisson processes
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by Fred W. Huffer, Jayaram Sethuraman and Sunder Sethuraman PDF

Proc. Amer. Math. Soc. 137 (2009), 2125-2134 Request permission

Abstract:

A sequence of random variables, each taking values $0$ or $1$, is called a Bernoulli sequence. 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 paper, 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. In particular, this general class includes all Bernoulli sequences considered in the literature, as well as a host of new sequences.

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