A study of counts of Bernoulli strings via conditional Poisson processes. (English) Zbl 1165.60305

Summary: 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.


60C05 Combinatorial probability
60K99 Special processes
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