A study of counts of Bernoulli strings via conditional Poisson processes

Abstract

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.

A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

“…are independent Poisson with means as above. This agrees with results in Holst (2007) and Huffer et al (2008).…”

“…With Z exponential with mean 1 and independent of P , which is Beta(a + 1, b − 1), we find that P = P e −Z/a is Beta(a, b). Using P , we can generate, by the embedding, a sequence Huffer et al (2008).…”

“…Inspired by Huffer et al (2008) we construct in this note an embedding in a marked Poisson process of a sequence of independent Bernoulli random variables with success probabilities a/(a +b+k −1), k = 1, 2, 3, . .…”

“…Other methods have previously been used to obtain such limits; see Arratia et al (2003), Holst (2007), Holst (2008), Huffer et al (2008), and the references therein. The embedding technique gives much more concise and transparent derivations and a better understanding of why the Poisson limits occur in such cases.…”

On Consecutive Records in Certain Bernoulli Sequences

“…Letting n → ∞ gives the limit distribution of Móri. This is also proved by other methods in Holst (2007), Holst (2008b), and Huffer et al (2009).…”