Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-08-09793-1
Published electronically: December 30, 2008
MathSciNet review: 2480294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60C05, 60K99

Retrieve articles in all journals with MSC (2000): 60C05, 60K99


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.

American Mathematical Society