Domains of Semi-Stable Attraction of Nonnormal Semi-Stable Laws

A sequence of independent, identically distributed random vectors X1, X2, ... is said to belong to the D-normed domain of semi-stable attraction of a random vector Y if there exist diagonal matrices An, constant vectors bn and a sequence (kn)n of natural numbers with kn ⇑ ∞ and kn+1/kn → c ≥ 1 such that An(X1 + · · · + Xkn) + bn converges in distribution to Y. The limit law Y is then called semi-stable. We present a simple, necessary, and sufficient condition for the existence of such An, bn, and kn in the case where Y has no normal component. Furthermore we prove some moment conditions for random vectors belonging to the D-normed domain of semi-stable attraction of Y.