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Poisson and compound Poisson approximations for random sums of random variables

Part of:Distribution theoryLimit theoremsStochastic processesDistribution theory - Probability

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.
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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 theorems62E17: Approximations to distributions (nonasymptotic)
Secondary:60G50: Sums of independent random variables; random walks60E05: Distributions62E20: Asymptotic distribution theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996, pp. 127 - 137
Copyright
Copyright © Applied Probability Trust 1996 

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