Journal of Applied Probability

Joint distributions of counts of strings in finite Bernoulli sequences

Fred W. Huffer and Jayaram Sethuraman

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

An infinite sequence (Y1, Y2,...) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,..., where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,... of occurrences of patterns or strings of the form {11}, {101}, {1001},..., respectively, in this sequence. The joint distribution of the counts Z1, Z2,... in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,..., Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

Article information

Source
J. Appl. Probab., Volume 49, Number 3 (2012), 758-772.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1346955332

Digital Object Identifier
doi:10.1239/jap/1346955332

Mathematical Reviews number (MathSciNet)
MR3012098

Zentralblatt MATH identifier
1314.60034

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60K99: None of the above, but in this section

Keywords
Conditional marked Poisson process Bernoulli sequence counts of strings random permutation cycles flaws and failures

Citation

Huffer, Fred W.; Sethuraman, Jayaram. Joint distributions of counts of strings in finite Bernoulli sequences. J. Appl. Probab. 49 (2012), no. 3, 758--772. doi:10.1239/jap/1346955332. https://projecteuclid.org/euclid.jap/1346955332


Export citation

References

  • Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519–535.
  • Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.
  • Hahlin, L.-O. (1995). Double records. Res. Rep. 1995:12, Department of Mathematics, Uppsala University.
  • Holst, L. (2007). Counts of failure strings in certain Bernoulli sequences. J. Appl. Prob. 44, 824–830.
  • Holst, L. (2008a). A note on embedding certain Bernoulli sequences in marked Poisson processes. J. Appl. Prob. 45, 1181–1185.
  • Holst, L. (2008b). The number of two consecutive successes in a Hoppe–Pólya urn. J. Appl. Prob. 45, 901–906.
  • Holst, L. (2009). On consecutive records in certain Bernoulli sequences. J. Appl. Prob. 46, 1201–1208.
  • Holst, L. (2011). A note on records in a random sequence. Ark. Mat. 49, 351–356.
  • Huffer, F., Sethuraman, J. and Sethuraman, S. (2008). A study of counts of Bernoulli strings via conditional Poisson processes. Preprint. Available at http://arxiv.org/abs/0801.2115v1.
  • Huffer, F., Sethuraman, J. and Sethuraman, S. (2009). A study of counts of Bernoulli strings via conditional Poisson processes. Proc. Amer. Math. Soc. 137, 2125–2134.
  • Joffe, A., Marchand, É., Perron, F. and Popadiuk, P. (2004). On sums of products of Bernoulli variables and random permutations. J. Theoret. Prob. 17, 285–292.
  • Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.
  • Kolchin, V. F. (1971). A problem of the allocation of particles in cells and cycles of random permutations. Theory Prob. Appl. 16, 74–90.
  • Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.
  • Sethuraman, J. and Sethuraman, S. (2004). On counts of Bernoulli strings and connections to rank orders and random permutations. In A festschrift for Herman Rubin (IMS Lecture Notes Monogr. Ser. 45), Institute for Mathematical Statistics, Beachwood, OH, pp. 140–152. \endharvreferences