Abstract
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.