SPECTRAL DECOMPOSITION FOR GENERALIZED DOMAINS OF SEMISTABLE ATTRACTION

Suppose X, X(1), X(2), X(3),... are i.i.d. random vectors, and k(n) a sequence of positive integers tending to infinity in such a way that k (n+1)/k(n)-->c greater than or equal to 1. If there exist linear opera tors A(n) and constant vectors b(n) such that A(n)(X(1) +...+ X(kn)) - b(n) converges in law to some full limit, then we say that the distri bution of X belongs to the generalized domain of semistable attraction of that limit law. The main result of this paper is a decomposition t heorem for the norming operators A(n), which allows its to reduce the problem to the case where the tail behavior of the limit law is essent ially uniform in all radial directions. Applications include a complet e description of moments, tails, centering, and convergence criteria.