Joint distributions of counts of strings in finite Bernoulli sequences

Abstract

An infinite sequence (Y₁, Y₂,...) of independent Bernoulli random variables with P(Y_i = 1) = a / (a + b + i - 1), i = 1, 2,..., where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z₁, Z₂, Z₃,... of occurrences of patterns or strings of the form {11}, {101}, {1001},..., respectively, in this sequence. The joint distribution of the counts Z₁, Z₂,... in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y₁, Y₂,..., Y_n) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z₁, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.