ON THE ASYMPTOTIC-BEHAVIOR OF SUMS OF MULTIDIMENSIONALLY INDEXED SEMI-STABLE RANDOM-VARIABLES

Let (X(n-->), neZ+d-->) be a sequence of identically distributed random variables with semi-regularly varying tails. We assume, whenever the random variables are integrable, that the mean is zero. As in the one dimensional case we are able to show that all normalized partial sums of weighted random variables cannot converge to one (a.s.) whenever our random variables have either a (1 + epsilon) moment or, in the non-L1 case, they fail to possess a (1 - epsilon) moment, for some epsilon > 0. The additional assumptions in this extension of the Z+1 case are extremely mild. Moreover, the conclusions hold for almost any partition of Z(+d).