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Journal of Applied Mathematics and Stochastic Analysis
Volume 15 (2002), Issue 3, Pages 219-233

New continuity estimates of geometric sums

Universidad Autónoma Metropolitana, Unidad Iztapalapa Av. Michaocán y la Purisima s/n, Col. Vicentina, Apartado Postal 55-534, México 09340, D.F., Mexico

Received 1 August 2000; Revised 1 March 2002

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper deals with sums of a random number of independent and identically distributed random variables. More specifically, we compare two such sums, which differ from each other in the distributions of their summands. New upper bounds (inequalities) for the uniform distance between distributions of sums are established. The right-hand sides of these inequalities are expressed in terms of Zolotarev's and the uniform distances between the distributions of summands. Such a feature makes it possible to consider these inequalities as continuity estimates and to apply them to the study of the stability (continuity) of various applied stochastic models involving geometric sums and their generalizations.