Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T17:49:45.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Part of: Combinatorial probability Special processes

Published online by Cambridge University Press:  04 February 2016

Fred W. Huffer  and
Jayaram Sethuraman
Fred W. Huffer*
Affiliation:
Florida State University
Jayaram Sethuraman*
Affiliation:
Florida State University
*
∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA.
∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Conditional marked Poisson processBernoulli sequencecounts of stringsrandom permutationcyclesflaws and failures

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 758 - 772
Copyright
© Applied Probability Trust 

References

Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519–535.Google Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.Google Scholar
Hahlin, L.-O. (1995). Double records. Res. Rep. 1995:12, Department of Mathematics, Uppsala University.Google Scholar
Holst, L. (2007). Counts of failure strings in certain Bernoulli sequences. J. Appl. Prob. 44, 824–830.CrossRefGoogle Scholar
Holst, L. (2008a). A note on embedding certain Bernoulli sequences in marked Poisson processes. J. Appl. Prob. 45, 1181–1185.CrossRefGoogle Scholar
Holst, L. (2008b). The number of two consecutive successes in a Hoppe–Pólya urn. J. Appl. Prob. 45, 901–906.Google Scholar
Holst, L. (2009). On consecutive records in certain Bernoulli sequences. J. Appl. Prob. 46, 1201–1208.Google Scholar
Holst, L. (2011). A note on records in a random sequence. Ark. Mat. 49, 351–356.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Kolchin, V. F. (1971). A problem of the allocation of particles in cells and cycles of random permutations. Theory Prob. Appl. 16, 74–90.Google Scholar
Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.Google Scholar
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.Google Scholar