  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.
