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Stein's method for geometric approximation

Part of: Combinatorial probability Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Erol A. Peköz
Erol A. Peköz*
Affiliation:
University of California, Los Angeles
*
∗Postal address: Department of Mathematics, University of California, Los Angeles, CA 90024, USA.
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Abstract

The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

Keywords

STEIN'S METHODGEOMETRIC APPROXIMATIONPOISSON APPROXIMATIONHITTING TIMESPATTERNS IN COIN TOSSING

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 707 - 713
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. Springer, Berlin.CrossRefGoogle Scholar
[2] Aldous, D. (1982) Markov chains with almost exponential hitting times. Stoch. Proc. Appl. 13, 305–310.CrossRefGoogle Scholar
[3] Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992) Compond Poisson approximation for non-negative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
[4] Barbour, A. D. and Grübel, R. (1995) The first divisible sum. J. Theoret. Prob. 8, 39–47.CrossRefGoogle Scholar
[5] Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press, Oxford.CrossRefGoogle Scholar
[6] Blom, G. and Thorburn, D. (1982) How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518–531.CrossRefGoogle Scholar
[7] Brown, M. and Ge, G. (1984) On the waiting time for the first occurrence of a pattern. In Reliability Theory and Models , ed. Abdel-Hameed, M., Çlinlar, E. and Quinn, J. CrossRefGoogle Scholar
[8] Chryssaphinou, O. and Papastavridis, S. (1988) A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials. J. Appl. Prob. 25, 428–431.CrossRefGoogle Scholar
[9] Ehm, W. (1991) Binomial approximation to the Poisson binomial distribution. Statist. Prob. Lett. 11, 7–16.CrossRefGoogle Scholar
[10] Godbole, A. P. (1991) Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851–865.CrossRefGoogle Scholar
[11] Godbole, A. P. and Schaffner, A. A. (1993) Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334–347.CrossRefGoogle Scholar
[12] Griffeath, D. (1976) Partial coupling and loss of memory for Markov chains. Ann. Prob. 4, 550–558.CrossRefGoogle Scholar
[13] Keilson, J. (1979) Markov Chain Models – Rarity and Exponentiality. Springer, Berlin.CrossRefGoogle Scholar
[14] Loh, W.-L. (1992) Stein's method and multinomial approximation. Ann. Appl. Prob. 2, 536–554.CrossRefGoogle Scholar
[15] Peköz, E. and Ross, S. M. (1994) Improving Poisson approximations. Prob. Eng. Inf. Sci. 8, 449–462.CrossRefGoogle Scholar
[16] Peköz, E. and Ross, S. M. (1995) A simple derivation of exact reliability formulas for linear and circular consecutive-k-of-n:F systems. J. Appl. Prob. 32, 554–557.CrossRefGoogle Scholar
[17] Pitman, J. W. (1976) On coupling of Markov chains. Z. Wahrscheinlichkeitsth. 35, 315–322.CrossRefGoogle Scholar