DOMAINS OF SEMI-STABLE ATTRACTION OF NONNORMAL SEMI-STABLE LAWS

A sequence of independent, identically distributed random vectors X(1) , X(2), ... is said to belong to the D-normed domain of semi-stable at traction of a random vector Y if there exist diagonal matrices A(n), c onstant vectors b(n) and a sequence (k(n))(n) of natural numbers with k(n) up arrow infinity and k(n+1)/k(n) --> c greater than or equal to 1 such that A(n)(X(1) + ... + X(kn)) + b(n) 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 A(n), b (n), and k(n) 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. (C) 1994 Academic Press , Inc.