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Poisson approximation for (k 1, k 2)-events via the Stein-Chen method

Part of: Limit theorems Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

P. Vellaisamy
P. Vellaisamy*
Affiliation:
Indian Institute of Technology, Bombay
*
∗ Postal address: Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India. Email address: [email protected]
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Abstract

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k 1, k 2) denote the number of times that k 1 failures are followed by k 2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k 1, k 2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for N k 3 (n; k 1, k 2), the number of times that k 1 failures followed by k 2 successes occur k 3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Keywords

(k 1, k 2)-eventbinomial distribution of order (k 1, k 2)Poisson approximationStein-Chen methodMarkov-Bernoulli variablerate of convergencelimit theorem

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic) 62E20: Asymptotic distribution theory
Secondary: 60E05: Distributions 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1081 - 1092
Copyright
Copyright © Applied Probability Trust 2004 

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References

Aki, S. (1985). Discrete distributions of order k on a binary sequence. Ann. Inst. Statist. Math. 37, 205–224.CrossRefGoogle Scholar
Aki, S., Kuboki, H., and Hirano, K. (1984). On discrete distributions of order k . Ann. Inst. Statist. Math. 36, 431–440.Google Scholar
Arratia, R., Goldstein, L., and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Barbour, A. D., and Eagleson, G. K. (1984). Poisson convergence for dissociated statistics. J. R. Statist. Soc. B 46, 397–402.Google Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Bartlett, M. S. (1978). An Introduction to Stochastic Processes, 3rd edn. Cambridge University Press.Google Scholar
Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851–865.CrossRefGoogle Scholar
Godbole, A. P. (1993). Approximate reliabilities of m-consecutive k-out-of-n: failure systems. Statistica Sinica 25, 321–327.Google Scholar
Godbole, A. P., and Schaffner, A. A. (1993). Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334–347.Google Scholar
Hirano, K. (1986). Some properties of the distributions of order k . In Fibonacci Numbers and Their Applications (Patras, 1984; Math. Appl. 28), eds Philippou, A. N. and Horadam, A. F., Reidel, Dordrecht, pp. 43–53.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
Ling, K. D. (1988). On binomial distributions of order k . Statist. Prob. Lett. 6, 247–250.Google Scholar
McGinley, W. G., and Sibson, R. (1975). Dissociated random variables. Math. Proc. Camb. Phil. Soc. 77, 185–188.Google Scholar
Philippou, A. N., and Makri, A. (1986). Success, runs and longest runs. Statist. Prob. Lett. 4, 211–215.Google Scholar
Philippou, A. N., Georghiou, C., and Philippou, G. N. (1983). A generalized distribution and some of its properties. Statist. Prob. Lett. 1, 171–175.Google Scholar
Yannaros, N. (1991). Poisson approximation for random sums of Bernoulli random variables. Statist. Prob. Lett. 11, 161–165.Google Scholar