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On the distance between the distributions of random sums

Part of: Limit theorems Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Bero Roos  and
Dietmar Pfeifer
Bero Roos*
Affiliation:
Universität Hamburg
Dietmar Pfeifer*
Affiliation:
Universität Oldenburg
*
∗ Postal address: Fachbereich Mathematik, SPST, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. Email address: roos@math.uni-hamburg.de
∗∗ Postal address: Fachbereich Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany.
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Abstract

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

Keywords

Binomial distributiondiscrete Taylor formulafinite signed measurePoisson approximationrandom sumsharpness resultsstochastic orderstop-loss ordertotal variation distance

MSC classification

Primary: 60E05: Distributions
Secondary: 62E17: Approximations to distributions (nonasymptotic) 60E15: Inequalities; stochastic orderings 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 87 - 106
Copyright
Copyright © Applied Probability Trust 2003 

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