Extension of the variance function of a steep exponential family

Let F={P(m,F); m∈MF} be a multidimensional steep natural exponential family parameterized by its domain of the means MF and let VF(m) be its variance function. This paper studies the boundary behaviour of VF. Necessary and sufficient conditions on a point m of ∂MF are given so that VF admits a continuous extension VF(m) to the point m . It is also shown that the existence of VF(m) implies the existence of a limit distribution P(m,F) concentrated on an exposed face of MF containing m. The relation between VF(m) and P(m,F) is established and some illustrating examples are given.