Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T11:19:56.605Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson and compound Poisson approximations for random sums of random variables

Part of: Distribution theory - Probability Stochastic processes Distribution theory Limit theorems

Published online by Cambridge University Press:  14 July 2016

P. Vellaisamy  and
B. Chaudhuri
P. Vellaisamy*
Affiliation:
Indian Institute of Technology
B. Chaudhuri*
Affiliation:
Indian Institute of Technology
*
∗Postal address for both authors: Department of Mathematics, Indian Institute of Technology, Powai, Bombay-400076, India.
∗Postal address for both authors: Department of Mathematics, Indian Institute of Technology, Powai, Bombay-400076, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

Keywords

RANDOM SUMSPOISSON AND COMPOUND POISSON APPROXIMATIONSCLUSTER BINOMIAL AND MULTINOMIAL DISTRIBUTIONS

MSC classification

Primary: 60F05: Central limit and other weak theorems 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60G50: Sums of independent random variables; random walks 60E05: Distributions 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 127 - 137
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.)

References

Arratia, R., Goldstein, L. and Gordon, L. (1990) Poisson approximation and the Chen-Stein method. Statist. Sci. 5, 403–434.Google Scholar
Barbour, A. D., Chen, L. H. Y. and Loh, Wei-Liem (1992) Compound Poisson approximation for non-negative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
Barbour, A. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Barbour, A. D. and Holst, L. (1989) Some applications of the Stein-Chen method for proving Poisson convergence. Adv. Appl. Prob. 21, 74–90.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press, Oxford.CrossRefGoogle Scholar
Brown, T. C. (1983) Some Poisson approximations using compensators. Ann. Prob 11, 726–744.Google Scholar
Chen, L. H. Y. (1975) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Finkelstein, M., Tucker, H. G. and Veeh, J. A. (1990) The limit distribution of the number of rare mutants. J. Appl. Prob. 27, 239–250.Google Scholar
Logunov, P. L. (1990) Estimates for the convergence rate to the Poisson distribution for random sums of independent indicators. Theory Prob. Appl. 35, 587–590.CrossRefGoogle Scholar
Loeve, M. (1955) Probability Theory. Van Nostrand, Princeton, NJ.Google Scholar
Panaretos, J. and Xekalaki, E. (1986) On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions. Ann. Inst. Statist. Math. A38, 223–231.CrossRefGoogle 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.Google Scholar
Serfozo, R. F. (1986) Compound Poisson approximations for sums of random variables. Ann. Prob. 14, 1391–1398.Google Scholar
Wang, Y. H. (1986) Coupling methods in approximations. Can. J. Statist. 14, 69–74.Google Scholar
Yannaros, N. (1991) Poisson approximation for random sums of Bernoulli random variables. Statist. Prob. Lett. 11, 161–165.Google Scholar