On Complete Convergence of the Sum of a Random Number of Random Variables

Abstract

Complete convergence of the normed random index sum of randomvariables of the form ${\sum }_{K=1}^{Tn}\hspace{0.17em}{X}_k\cancel{}\hspace{0.17em}Tn,\hspace{0.17em}\hspace{0.17em}n\hspace{0.17em}⩾\text{1}$ is established. While the random variables {X_n, n ⩾ 1} are i.i.d., no assumptions are placed on the joint distribution of the indices {T_n, n ⩾ 1}.