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.