Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T18:44:31.675Z Has data issue: false hasContentIssue false
  • English
  • Français

On improvements of the order of approximation in the Poisson limit theorem

Part of: Limit theorems Groups and semigroups of linear operators, their generalizations and applications Classical measure theory

Published online by Cambridge University Press:  14 July 2016

K. Borovkov  and
D. Pfeifer
K. Borovkov*
Affiliation:
University of Melbourne
D. Pfeifer*
Affiliation:
Universität Hamburg
*
∗Postal address: Department of Statistics, University of Melbourne, Parkville 3052, Australia.
∗∗Postal address: Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

Keywords

SIGNED MEASURESRECORDSOPERATOR SEMIGROUPS

MSC classification

Primary: 60F05: Central limit and other weak theorems 28A33: Spaces of measures, convergence of measures
Secondary: 47D03: Groups and semigroups of linear operators
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 146 - 155
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Footnotes

This research was partially supported by a grant of the Alexander von Humboldt Foundation, Germany.

References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, D.C. Google Scholar
Borovkov, K. A. (1988) Refinement of Poisson approximation. Theory Prob. Appl. 33, 343–347.CrossRefGoogle Scholar
Borovkov, K. and Pfeifer, D. (1993) On record indices and record times. J. Statist. Plann. Inference 45, 65–79.CrossRefGoogle Scholar
Butzer, P. L., Hauss, M. and Schmidt, M. (1989) Factorial functions and Stirling numbers of fractional orders. Res. Math. 16, 16–48.CrossRefGoogle Scholar
Charalambides, Ch. A. and Singh, J. (1988) A review of the Stirling numbers, their generalizations and statistical applications. Commun. Statist. Theor. Meth. 17, 2533–2592.Google Scholar
Deheuvels, P. and Pfeifer, D. (1986) A semigroup approach to Poisson approximation. Ann. Prob. 14, 665–678.Google Scholar
Deheuvels, P. and Pfeifer, D. (1987) Semigroups and Poisson approximation. In New Perspectives in Theoretical and Applied Statistics. pp, 439–448. Wiley, New York.Google Scholar
Deheuvels, P. and Pfeifer, D. (1988) On a relationship between Uspensky's theorem and Poisson approximations. Ann. Inst. Statist. Math. 40, 671–681.Google Scholar
Deheuvels, P., Pfeifer, D. and Puri, M. L. (1989) A new semigroup technique in Poisson approximation. Semigroup Forum 38, 198–201.Google Scholar
Dwass, M. (1960) Some k-sample rank order tests. In: Contributions to Probability and Statistics: Essays in Honor of H. Hotelling. ed. Olkin, I. et al. pp. 198–202. Stanford University Press, Stanford, CA.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980) Tables of Series, Products, and Integrals. Academic Press, New York.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Discrete Distributions. Wiley, New York.Google Scholar
Jordan, C. (1965) Calculus of Finite Differences. 3rd edn. Chelsea, New York.Google Scholar
Kruopis, J. (1986) Precision of approximation of the generalized binomial distribution by convolutions of Poisson measures. Lithuanian Math. J. 26, 37–49.Google Scholar
Kemp, R. (1984) Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley-Teubner, New York.Google Scholar
Nevzorov, V. B. (1988) Records. Theory Prob. Appl. 32, 201–228.CrossRefGoogle Scholar
Pfeifer, D. (1989) Extremal processes, secretary problems and the 1/e-law. J. Appl. Prob. 26, 722–733.CrossRefGoogle Scholar
Pfeifer, D. (1991) some remarks on Nevzorov's record model. Adv. Appl. Prob. 23, 823–834.Google Scholar
Rényi, A. (1962) Théorie des éléments saillants d'une suite d'observations. Colloq. on Combinatorial Methods in Probability Theory. pp. 104–115. Mathematisk Institut, Aarhus Universität, Denmark.Google Scholar
Ross, S. M. (1982) A simple heuristic approach to simplex efficiency. Eur. J. Operat. Res. 9, 344–346.CrossRefGoogle Scholar
Shorgin, S. Y. (1977) Approximation of a generalized binomial distribution. Theory Prob. Appl. 22, 846–850.Google Scholar
Tzaregradskii, I. P. (1958) On a uniform approximation to the binomial distribution by infinitely divisible distributions. Theory Prob. Appl. 3, 434–438.Google Scholar