Almost Sure Approximation of the Superposition of the Random Processes

Abstract

This article presents sufficient conditions, which provide almost sure (a.s.) approximation of the superposition of the random processes S(N(t)), when cad-lag random processes S(t) and N(t) themselves admit a.s. approximation by a Wiener or stable Levy processes. Such results serve as a source of numerous strong limit theorems for the random sums under various assumptions on counting process N(t) and summands. As a consequence we obtain a number of results concerning the a.s. approximation of the Kesten–Spitzer random walk, accumulated workload input into queuing system, risk processes in the classical and renewal risk models with small and large claims and use such results for investigation the growth rate and fluctuations of the mentioned processes.