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Limit theorems for the number of occurrences of consecutive k successes in n Markovian trials

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Y. H. Wang  and
Shuixin Ji
Y. H. Wang*
Affiliation:
Concordia University
Shuixin Ji*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd W, Montréal, PQ Canada H3G 1M8.
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd W, Montréal, PQ Canada H3G 1M8.
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Abstract

We present a method of deriving the limiting distributions of the number of occurrences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string of S of exact length k preceded and followed by an F, except the first run which may not be preceded by an F, or the last run which may not be followed by an F. 2. A string of S of length k or more. 3. A string of S of exact length k, where recounting starts immediately after a run occurs. 4. A string of S of exact length k, allowing overlapping runs. It is shown that the limits are convolutions of two or more distributions with one of them being either Poisson or compound Poisson, depending on the type of runs in question. The completely stationary Markov case and the i.i.d. case are also treated.

Keywords

RUNSMARKOV CHAINSSTATIONARY TRANSITION PROBABILITIESCONVERGENCEPOISSONCOMPOUND POISSONRELIABILITYCONSECUTIVE-k-OUT-OF n: F SYSTEM

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 727 - 735
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The first author's research is partially supported by the Natural Sciences and Engineering Council of Canada, #OGPIN 014.

References

Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Clarendon Press, Oxford.Google Scholar
Bühler, W. J. (1989) Probabilistic derivation for the limit of the Markov binomial distributions. Math. Scientist. 14, 117–119.Google Scholar
Chao, M. T. and Fu, J. C. (1989) A limit theorem of certain repairable systems. Ann. Inst. Statist. Math. 41, 809–818.Google Scholar
Dobrušin, R. L. (1953) Limit theorems for a Markov chain of two states. Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 17, 291–330.Google Scholar
Doeblin, W. (1938) Exposé de la théorie des chaines simples constantes de Markoff à un nombre fini d'états. Rev. Math, de l'Union Interbalkanique 2, 77–105.Google Scholar
Feller, W. (1950) An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, New York.Google Scholar
Fu, J. C. (1993) Poisson convergence in reliability of a large connected system as related to coin tossing. Statistica Sinica 3, 261–276.Google Scholar
Gani, J. (1982) On the probability generating function of the sum of Markov Bernoulli random variables. J. Appl. Prob. 19A, 321–326.Google Scholar
Godbole, A. P. (1990) Degenerate and Poisson convergence criteria for success runs. Statist. Prob. Lett. 10, 119–124.CrossRefGoogle Scholar
Goldstein, L. (1990) Poisson approximation and DNA sequence matching, Comm. Statist.-Theory Meth. 19, 4167–4179.Google Scholar
Hirano, K. (1984) Some properties of the distributions of order k. Proc. 1st Internat. Conf. Fibonacci Numbers and Their Applications, ed. Philippou, A. N. and Horadam, A. F. Gutenberg, Athens.Google Scholar
Huang, W. T. and Tsai, C. S. (1991) On a modified binomial distribution of order k. Statist. Prob. Lett. 11, 125–131.Google Scholar
Koopman, B. O. (1950) A generalization of Poisson's distribution for Markoff chains. Proc. Nat. Acad. Sci. USA 36, 202–207.Google Scholar
Koutras, M. V. and Papastavridis, S. G. (1993) On the number of runs and related statistics. Statistica Sinica. 3, 277–294.Google Scholar
Ling, K. D. (1988) On binomial distributions of order k. Statist. Prob. Lett. 6, 247–250.Google Scholar
Mood, A. (1940) The distribution theory of runs. Ann. Math. Statist, 11, 367–392.Google Scholar
Papastavridis, S. (1987) A limit theorem for reliability of a consecutive-k-out-of-n:F system. Adv. Appl. Prob. 19, 746–748.Google Scholar
Pedler, J. (1978) The occupation time, number of transitions, and waiting time for two-state Markov chains. , The Flinders University of South Australia.Google Scholar
Wang, Y. H. (1981) On the limit of the Markov binomial distribution. J. Appl. Prob. 18, 937–942.CrossRefGoogle Scholar
Wang, Y. H. (1989) From Poisson to compound Poisson approximations. Math. Scientist. 14, 38–49.Google Scholar
Wang, Y. H. (1992) Approximating the kth order two-state Markov Bernoulli chains. J. Appl. Prob. 29, 861–868.CrossRefGoogle 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 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