Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T22:06:24.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence theorems for the lengths of consecutive successes of Markov Bernoulli sequences

Part of: Distribution theory

Published online by Cambridge University Press:  14 July 2016

Y. H. Wang ,
Hsiao-Fen Chang  and
Shun-Yi Chen
Y. H. Wang*
Affiliation:
Tunghai University
Hsiao-Fen Chang*
Affiliation:
Tamkang University
Shun-Yi Chen*
Affiliation:
Tamkang University
*
∗Postal address: Department of Statistics, Tunghai University, Taichung, Taiwan, 407. Email address: yhwang@mail.thu.edu.tw
∗∗Postal address: Department of Mathematics, Tamkang University, Taipei, Taiwan, 100
∗∗Postal address: Department of Mathematics, Tamkang University, Taipei, Taiwan, 100
Get access Rights & Permissions [Opens in a new window]

Abstract

The limiting distributions of the sums of the lengths of four different kinds of runs of consecutive successes of Markov Bernoulli sequences are derived. It is shown that the limits are convolutions of several distributions involving the Bernoulli, the geometric and the Poisson or compound Poisson distributions.

Keywords

Limit theoremPoisson limitconsecutive successes of length ksum of random variablesreliability

MSC classification

Primary: 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 741 - 749
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balakrishnan, N., and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Chow, Y. S. (1992). On a Poisson convergence for the number of runs. J. Chinese Statist. Assoc. 30, 241–244 (in Chinese).Google Scholar
Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Vol. 1. John Wiley, New York.Google Scholar
Fu, J. C., and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.Google Scholar
Wang, Y. H. and Bühler, W. J. (1991). Renewal process proof for the limit of the Markov binomial distribution. Math. Scientist 16, 66–68.Google Scholar
Wang, Y. H., and Ji, S. X. (1995). Limit theorems for the number of occurrences of consecutive k successes in n Markov Bernoulli trials. J. Appl. Prob. 32, 727–735.CrossRefGoogle Scholar
Wang, Y. H., and Liang, Z. Y. (1993). The probability of occurrences of runs of length k in n Markov Bernoulli trials. Math. Scientist 18, 105–112.Google Scholar