On Tails and Domains of Attraction of Stable Measures in Banach Spaces

The exact tail behavior of stable measures in Banach spaces and measures in their domains of attraction is given.Conditions for a p.m. to be in the domain of attraction of a stable p.m. of order a are derived which are sufficient in type/ spaces,/) > a, and necessary in general.This paper also contains a short proof of the Lvy-Khinchin formula in Banach spaces.1. Introduction.Let B he a separable Banach space and p a stable law on (B, <$> ), i.e. p is such that if A, are i.i.d. with t(X) = p, then for every n there exist a" and b" such that (27=1A,/a" -b") = t(X).It is well known that the only possible values for an are nx/a, a G (0, 2].We then call a the index or order of p.A probability measure X belongs to the domain of attraction of p if and only if there exist {an} c R and {bn} c B such that whenever X are i.i.d. with (A",) = X, then t(^n=xX/an -b")^>wp.It is also well known that only stable measures have nonvoid domains of attraction (Kumar and Mandrekar (1972)).In case the constants an can be chosen to be an = '/", a the order of p, then X is in domain of normal attraction of p.The tail behavior of stable measures and measures in their domain of attraction plays an important role in the study of domains of attraction on the line (see e.g.Feller (1970)) and in other spaces as well (Kuelbs and Mandrekar (1974) among others).The tail behavior of a stable p.m. is in turn completely determined by its associated Levy measure.In this paper we present results on these subjects.In 2 we give a proof of the Lvy-Khinchin representation of infinitely divisible laws in Banach space.Proofs of this theorem have been given independently by Araujo (1975) andDettweiler (1976).