Domains of attraction for exponential families

With the df F of the rv X we associate the natural exponential family of df's F_λ where dF_λ (x)=e^λx dF(x)/Ee^λX for λ∈Λ:={λ∈ℝ Ee^λX<∞}. Assume λ_∞=sup Λ≤∞ does not lie in Λ. Let λ↑λ_∞, then non-degenerate limit laws for the normalised distributions F_λ(a_λx+b_λ) are the normal and gamma distributions. Their domains of attractions are determined. Applications to saddlepoint and gamma approximations are considered.