Domains of attraction for exponential families

With the df F of the rv X we associate the natural exponential family of dfʹs Fλ wheredFλ(x)=eλx dF(x)/EeλXfor λ∈Λ≔{λ∈R | EeλX<∞}. Assume λ∞=sup Λ⩽∞ does not lie in Λ. Let λ↑λ∞, then non-degenerate limit laws for the normalised distributions Fλ(aλx+bλ) are the normal and gamma distributions. Their domains of attractions are determined. Applications to saddlepoint and gamma approximations are considered.