Abelian theorems for a class of probability distributions inR d and their application

Abelian Theorems, Limit Properties of Conjugate Distributions, and Large Deviations for Sums of Independent Random Vectors

“…Abel-type theorem. Conditions (A), (B), and (C) are the uniform analogues of the corresponding conditions in [19]. Therefore, the statement of Theorem 1.3 from that paper holds uniformly in e ∈ S d−1 .…”

“…Repeating arguments leading to the assertion of Theorem 1.3 in [19] and Theorem 3.1, one can show that if q(α − 1) + (d + 1)/2 > 0 and τ ↓ 0, then…”

“…The shape of the set of existence for the moment generating function f (s) = R d e s,x p(x) dx was established in [19]. It turns out that if p(x) has the representation (2.1), where a(x) ∈ A and b(x) is β-regularly varying as |x| → ∞, then the moment generating function f (s) is bounded in int S, where S is a bounded convex set containing the origin and such that a(x) is its support function.…”

“…Following the proof of Theorem 1.3 in [19] and Theorem 3.1, it is not difficult to show that as s = (h(e) − σ(e) τ ) e, τ ↓ 0, for all i = 1, . .…”

“…These questions are of independent interest at least within the framework of asymptotic analysis. The answers concerning the considered class of the underlying distributions were given in [19]. They play an important role in the considerations presented below.…”