Two-Sided Bounds of Random Sums with Subexponential Summands

Two-sided bounds of the distribution function of a sum of independent identically distributed positive random variables are derived in the case where the number of summands is random, independent of the values of the summands, and has a discrete distribution of a general form. Such random sums model important characteristics of communication networks, queueing systems, reliability models, and risk models. The basic goal of the work is deriving asymptotically correct upper and lower bounds in the case of subexponentially distributed summands. The bounds proposed are new.