Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T17:27:04.567Z Has data issue: false hasContentIssue false
  • English
  • Français

On Consecutive Records in Certain Bernoulli Sequences

Part of: Combinatorial probability Special processes

Published online by Cambridge University Press:  14 July 2016

Lars Holst
Lars Holst*
Affiliation:
Royal Institute of Technology
*
∗Postal address: Department of Mathematics, Royal Institute of Technology, SE–100 44 Stockholm, Sweden. Email address: [email protected]
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.

In an infinite sequence of independent Bernoulli trials with success probabilities pk=a/(a+b +k-1) for k=1,2,3,…, let Nr be the number of r≥2 consecutive successes. Expressions for the first two moments of Nr are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of Nr can be calculated for small r.

Keywords

Bernoulli trialembedding in Poisson processrandom permutationrecordsruns of successessums of indicators

MSC classification

Secondary: 60C05: Combinatorial probability 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1201 - 1208
Copyright
Copyright © Applied Probability Trust 2009 

References

Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519–535.CrossRefGoogle Scholar
Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.CrossRefGoogle Scholar
Chern, H.-H. and Hwang, H.-K. (2005). Limit distribution of the number of consecutive records. Random Structures Algorithms 26, 404–417.CrossRefGoogle Scholar
Chern, H.-H., Hwang, H.-K. and Yeh, Y.-N. (2000). Distribution of the number of consecutive records. Random Structures Algorithms 17, 169–196.3.0.CO;2-K>CrossRefGoogle 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.Google Scholar
Holst, L. (2008a). The number of two consecutive successes in a Hoppe–Pólya urn. J. Appl. Prob. 45, 901–906.Google Scholar
Holst, L. (2008b). A note on embedding certain Bernoulli sequences in marked Poisson processes. J. Appl. Prob. 45, 1181–1185.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.CrossRefGoogle Scholar
Móri, T. F. (2001). On the distribution of sums of overlapping products. Acta Sci. Math. 67, 833–841.Google Scholar