## Journal of Applied Probability

- J. Appl. Probab.
- Volume 49, Number 3 (2012), 758-772.

### Joint distributions of counts of strings in finite Bernoulli sequences

Fred W. Huffer and Jayaram Sethuraman

#### 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.

#### Article information

**Source**

J. Appl. Probab., Volume 49, Number 3 (2012), 758-772.

**Dates**

First available in Project Euclid: 6 September 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1346955332

**Digital Object Identifier**

doi:10.1239/jap/1346955332

**Mathematical Reviews number (MathSciNet)**

MR3012098

**Zentralblatt MATH identifier**

1314.60034

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60K99: None of the above, but in this section

**Keywords**

Conditional marked Poisson process Bernoulli sequence counts of strings random permutation cycles flaws and failures

#### Citation

Huffer, Fred W.; Sethuraman, Jayaram. Joint distributions of counts of strings in finite Bernoulli sequences. J. Appl. Probab. 49 (2012), no. 3, 758--772. doi:10.1239/jap/1346955332. https://projecteuclid.org/euclid.jap/1346955332