A study of counts of Bernoulli strings via conditional Poisson processes

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.