WEAK COMPACTNESS OF RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES.

The shift-compactness of random sums S[SUP(n),SUBN[SUBn]], S[SUP(n),SUBk] = X[SUBn,1] +…+ X[SUBn,k], of independent random variables is investigated under the assumptions that in each sum the summands and their number N[SUBn] are independent and that the summands satisfy the condition of uniform asymptotic negligibility in the form [This equation is not coverted in ascii text.] in probability for each ε > 0. Some necessary and sufficient conditions are given for the weak compactness of random sums S[SUP(n),SUBN[SUBn]] -- A[SUBn], and the form of centering constants A[SUBn] is described. [ABSTRACT FROM AUTHOR]