Mm. Meerschaert et Hp. Scheffler, SPECTRAL DECOMPOSITION FOR GENERALIZED DOMAINS OF SEMISTABLE ATTRACTION, Journal of theoretical probability, 10(1), 1997, pp. 51-71
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.