Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Abstract

An infinite sequence (Y 1 , Y 2 , . . . ) 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 Z 1 , Z 2 , Z 3 , . . . of occurrences of patterns or strings of the form {11}, {101}, {1001}, . . . , respectively, in this sequence. The joint distribution of the counts Z 1 , Z 2 , . . . in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1 , Y 2 , . . . , Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1 , 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.

On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

“…respectively. Some relevant contributions on the subject are the works of Sarkar et al [12], Sen and Goyal [13], Holst [4], Dafnis et al [14], Huffer and Sethuraman [15], and Makri and Psillakis [16]. Applications of constrained ( , ℓ) strings of the general ( ≤ ≤ ℓ) or the restricted forms ( ≤ ℓ, = , ≥ ) were found in information theory and data compression (see Zehavi and Wolf [17], Jacquet and Szpankowski [18], and Stefanov and Szpankowski [19]) in urn models, record models, and random permutations (see Chern et al [20], Joffe et al [21], Chern and Hwang [22], Holst [5,9,10,23], and Huffer et al [24]) in system reliability (see Eryilmaz and Zuo [25], Eryilmaz and Yalcin [6], and Makri [26]) and in biomedical engineering (see Dafnis and Philippou [27]).…”