On Random Sums of Random Vectors

Abstract

To obtain the limit distribution of a sequence $T_n$ of random vectors, the $j$th component of $T_n$ being the sum of a random number $N_n^{(j)}$ of $j$th components of independent, identically distributed chance vectors $X_n$, it is first necessary to treat the special case where the $N_n^{(j)}$ are degenerate random variables. This is done in Section 2 and generalized to infinitely divisible limits in Section 4. The basic problem is treated in Section 3 and generalizations of theorems of Doeblin [3], Anscombe [1] and Renyi [7] are obtained.