Conditional natural exponential families

Let F={Pθ;θ∈Θ}F={Pθ;θ∈Θ} be a natural exponential family on RdRd and let H be an exposed face of the closed convex hull of the F   support. The aim of this paper is to study the asymptotic behavior of the law Pθ+λuPθ+λu as λλ increases to +∞+∞, for all u exterior normal vector of H  . We obtain a degenerate weak limit Pθ,+∞Pθ,+∞ concentrated on the face H  . The relation between the conditional law of PθPθ given H   and Pθ,+∞Pθ,+∞ is established and some illustrating examples are given.