A study of counts of Bernoulli strings via conditional Poisson processes
HTML articles powered by AMS MathViewer
- by Fred W. Huffer, Jayaram Sethuraman and Sunder Sethuraman PDF
- Proc. Amer. Math. Soc. 137 (2009), 2125-2134 Request permission
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.
References
- Richard Arratia, A. D. Barbour, and Simon Tavaré, Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab. 2 (1992), no. 3, 519–535. MR 1177897
- Richard Arratia, A. D. Barbour, and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003. MR 2032426, DOI 10.4171/000
- Richard Arratia and Simon Tavaré, The cycle structure of random permutations, Ann. Probab. 20 (1992), no. 3, 1567–1591. MR 1175278
- Hua-Huai Chern, Hsien-Kuei Hwang, and Yeong-Nan Yeh, Distribution of the number of consecutive records, Proceedings of the Ninth International Conference “Random Structures and Algorithms” (Poznan, 1999), 2000, pp. 169–196. MR 1801131, DOI 10.1002/1098-2418(200010/12)17:3/4<169::AID-RSA1>3.0.CO;2-K
- W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc. 51 (1945), 800–832. MR 13252, DOI 10.1090/S0002-9904-1945-08448-1
- J. K. Ghosh and R. V. Ramamoorthi, Bayesian nonparametrics, Springer Series in Statistics, Springer-Verlag, New York, 2003. MR 1992245
- Lars Holst, Counts of failure strings in certain Bernoulli sequences, J. Appl. Probab. 44 (2007), no. 3, 824–830. MR 2355594, DOI 10.1239/jap/1189717547
- Anatole Joffe, Éric Marchand, François Perron, and Paul Popadiuk, On sums of products of Bernoulli variables and random permutations, J. Theoret. Probab. 17 (2004), no. 1, 285–292. MR 2054589, DOI 10.1023/B:JOTP.0000020485.34082.8c
- Kolchin, V.F. (1971), A problem of the allocation of particles in cells and cycles of random permutations. Theory Probab. Appl. 16 74-90.
- Ramesh M. Korwar and Myles Hollander, Contributions to the theory of Dirichlet processes, Ann. Probability 1 (1973), 705–711. MR 350950, DOI 10.1214/aop/1176996898
- Tamás F. Móri, On the distribution of sums of overlapping products, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 833–841. MR 1876470
- Sidney Resnick, Adventures in stochastic processes, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1181423
- Jayaram Sethuraman and Sunder Sethuraman, On counts of Bernoulli strings and connections to rank orders and random permutations, A festschrift for Herman Rubin, IMS Lecture Notes Monogr. Ser., vol. 45, Inst. Math. Statist., Beachwood, OH, 2004, pp. 140–152. MR 2126893, DOI 10.1214/lnms/1196285386
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
- MR Author ID: 612250
- Email: sethuram@iastate.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2480294