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A study of counts of Bernoulli strings via conditional Poisson processes
Author(s):
Fred
W.
Huffer;
Jayaram
Sethuraman;
Sunder
Sethuraman
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2125-2134.
MSC (2000):
Primary 60C05;
Secondary 60K99
Posted:
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:
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
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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