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), 21252134
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 , 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:
http://dx.doi.org/10.1090/S0002993908097931
PII:
S 00029939(08)097931
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 AROW911NF0410333, NSAH982300510041, and NSFDMS0504193.
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.
