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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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.


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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: http://dx.doi.org/10.1090/S0002-9939-08-09793-1
PII: S 0002-9939(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.