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A unification of some approaches to Poisson approximation

Published online by Cambridge University Press:  14 July 2016

Hans-Jürgen Witte
Hans-Jürgen Witte*
Affiliation:
University of Oldenburg
*
∗Postal address: Universität Oldenburg, Fachbereich 6 Mathematik, Postfach 2503, D-2900 Oldenburg, W. Germany.
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Abstract

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.

Keywords

TOTAL VARIATION DISTANCEKOLMOGOROV METRICFORTET–MOURIER METRICGENERATING FUNCTIONSSEMIGROUPS
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 611 - 621
Copyright
Copyright © Applied Probability Trust 1990 

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References

Barbour, A. D. (1987) Asymptotic expansions in the Poisson limit theorem. Ann. Prob. 15, 748–766.CrossRefGoogle Scholar
Barbour, A. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.Google Scholar
Bruss, F. T. (1984) Patterns of relative maxima in random permutations Ann. Soc. Sci. Bruxelles 98 (I), 19–28.Google Scholar
Chen, L. H. Y. (1975) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Deheuvels, P. and Pfeifer, D. (1986a) A semigroup approach to Poisson approximation. Ann. Prob. 14, 665–678.Google Scholar
Deheuvels, P. and Pfeifer, D. (1986b) Operator semigroups and Poisson convergence in selected metrics. Semigroup Forum 34, 203–224.Google Scholar
Deheuvels, P. and Pfeifer, D. (1989) On a relationship between Uspensky's theorem and Poisson approximation. Inst. Math. Stat. Google Scholar
Deheuvels, P., Karr, A., Pfeifer, D. and Serfling, R. J. (1988) Poisson convergence in selected metrics by coupling and semigroup methods with applications. Statist. Plan. Inf. Google Scholar
Deheuvels, P., Pfeifer, D. and Puri, M. L. (1989) A new semigroup technique in Poisson approximation. Semigroup Forum 38, 189–201.Google Scholar
Franken, P. (1964) Approximation der Verteilungen von Summen unabhängiger Zufallsgrößen durch Poissonsche Verteilungen. Math. Nachr. 27, 303–340.Google Scholar
Fortet, R. and Mourier, E. (1953) Convergence de la repartition empirique vers la repartition théoretique. Ecole Normale Supérieur Ser. 3 70, 266–285.Google Scholar
Gastwirth, J. L. and Bhattacharya, P. K. (1984) Probability models of pyramid or chain letter systems demonstrating that their promotional claims are unreliable. Operat. Res. 32, 527–536.CrossRefGoogle Scholar
Kemp, R. (1984) Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley-Teubner, New York.Google Scholar
Kerstan, J. (1964) Verallgemeinerung eines Satzes von Prochorov und Le Cam. Z. Wahrscheinlichkeitsth. 2, 173–179.Google Scholar
Le Cam, L. (1960) An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197.Google Scholar
Pfeifer, D. (1989) Extremal processes, secretary problems and the 1/e-law. J. Appl. Prob. 26, 722–733.Google Scholar
Renyi, A. (1962) Théorie des elements saillants d'une suite d'observations. Coll. Comb. Meth. Prob. Th. 104–115, Mathematisk Institut, Aarhus Universitet, Denmark.Google Scholar
Ross, S. M. (1982) A simple heuristic approach to simplex efficiency. Eur. J. Operat. Res. 9, 344–346.Google Scholar
Serfling, R. J. (1975) A general Poisson approximation theorem. Ann. Prob. 3, 726–731.CrossRefGoogle Scholar
Serfling, R. J. (1978) Some elementary results on Poisson approximation in a sequence of Bernoulli trials. SIAM Rev. 20, 567–579.CrossRefGoogle Scholar
Shorgin, S. Y. A. (1977) Approximation of a generalized binomial distribution. Theory Prob. Appl. 22, 846–850.Google Scholar
Uspensky, J. V. (1931) On Ch. Jordan's series for probability. Ann. Math. 32, 306–312.Google Scholar
Vallender, S. S. (1973) Calculation of the Wasserstein distance between distributions on the line. Theory Prob. Appl. 18, 784–786.CrossRefGoogle Scholar
Zolotarev, V. M. (1976) Metric distances in spaces of random variables and their distributions. Math. USSR Sb. 30, 373–401.Google Scholar
Zolotarev, V. M. (1983) Probability metrics. Theory Prob. Appl. 28, 278–302.Google Scholar