A study of counts of Bernoulli strings via conditional Poisson processes

Authors:
Fred W. Huffer, Jayaram Sethuraman and Sunder Sethuraman

Journal:
Proc. Amer. Math. Soc. **137** (2009), 2125-2134

MSC (2000):
Primary 60C05; Secondary 60K99

Published electronically:
December 30, 2008

MathSciNet review:
2480294

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Abstract | References | Similar Articles | Additional Information

Abstract: A sequence of random variables, each taking values 0 or , is called a Bernoulli sequence. We say that a string of length occurs in a Bernoulli sequence if a success is followed by exactly failures before the next success. The counts of such -strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic -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 -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.

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Additional Information

**Fred W. Huffer**

Affiliation:
Department of Statistics, Florida State University, Tallahassee, Florida 32306

Email:
huffer@stat.fsu.edu

**Jayaram Sethuraman**

Affiliation:
Department of Statistics, Florida State University, Tallahassee, Florida 32306

Email:
sethu@stat.fsu.edu

**Sunder Sethuraman**

Affiliation:
Department of Mathematics, 396 Carver Hall, Iowa State University, Ames, Iowa 50011

Email:
sethuram@iastate.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09793-1

Keywords:
Bernoulli,
cycles,
strings,
spacings,
nonhomogeneous,
Poisson processes,
random permutations

Received by editor(s):
January 14, 2008

Received by editor(s) in revised form:
September 25, 2008

Published electronically:
December 30, 2008

Additional Notes:
This research was partially supported by ARO-W911NF-04-1-0333, NSA-H982300510041, and NSF-DMS-0504193.

Communicated by:
Edward C. Waymire

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.