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

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

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.

**1.**Arratia, R., Barbour, A.D., and Tavaré, S. (1992), Poisson process approximations for the Ewens sampling formula.*Ann. Appl. Probab.***2**519-535. MR**1177897 (94a:60003)****2.**Arratia, R., Barbour, A.D., and Tavaré, S. (2003),*Logarithmic Combinatorial Structures: A Probabilistic Approach.*European Mathematical Society, Zürich. MR**2032426 (2004m:60004)****3.**Arratia, R., and Tavaré, S. (1992), The cycle structure of random permutations.*Ann. Probab.***20**1567-1591. MR**1175278 (93g:60013)****4.**Chern, H.-H., Hwang, H.-K., and Yeh, Y.-N. (2000), Distribution of the number of consecutive records.*Random Structures and Algorithms***17**169-196. MR**1801131 (2002c:60006)****5.**Feller, W. (1945), The fundamental limit theorems in probability.*Bull. Amer. Math. Soc.***51**800-832. MR**0013252 (7:128i)****6.**Ghosh, J.K., and Ramamoorthi, R.V. (2003),*Bayesian Nonparametrics*, Springer-Verlag, New York. MR**1992245 (2004g:62004)****7.**Holst, Lars (2007), Counts of failure strings in certain Bernoulli sequences.*J. Appl. Probab.***44**824-830. MR**2355594 (2008i:60014)****8.**Joffe, A., Marchand, E., Perron, F., and Popadiuk, P. (2004), On sums of products of Bernoulli variables and random permutations.*Journal of Theoretical Probability***17**285-292. MR**2054589 (2005e:60023)****9.**Kolchin, V.F. (1971), A problem of the allocation of particles in cells and cycles of random permutations.*Theory Probab. Appl.***16**74-90.**10.**Korwar, R.M., and Hollander, M. (1973), Contributions to the theory of Dirichlet processes.*Ann. Probab.***1**705-711. MR**0350950 (50:3442)****11.**Móri, T.F. (2001), On the distribution of sums of overlapping products.*Acta Scientarium Mathematica (Szeged)***67**833-841. MR**1876470 (2002h:60024)****12.**Resnick, S.I. (1994),*Adventures in Stochastic Processes.*Second Ed., Birkhäuser, Boston. MR**1181423 (93m:60004)****13.**Sethuraman, Jayaram, and Sethuraman, Sunder (2004), On counts of Bernoulli strings and connections to rank orders and random permutations. In*A festschrift for Herman Rubin. IMS Lecture Notes Monograph Series***45**140-152. MR**2126893 (2006d:60020)**

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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.