Stability for multivariate exponential families (Q1600633)

Let \(E\) be a Euclidean space, let \(Z :\Omega\to E\) be a nondegenerate random vector, and suppose there is an open convex set \(D\subset E\) such that \(P(Z\in \overline{D}) = 1\). If \(\mu\) is the distribution of \(Z\), define measures \(\mu_\lambda\) by \(d\mu_\lambda(x) = e^{\lambda x}d\mu(x)\), \(x\in E\), for any \(\lambda\) in the dual space \(E^T\), and set \(\Lambda = \{\lambda\in E^T: \mu_\lambda(E) < \infty\}\). For \(\lambda\in\Lambda\), consider a random vector \(Z_\lambda\) with probability distribution \(\mu_\lambda/\mu_\lambda(E)\). The family \(\{Z_\lambda,\;\lambda\in\Lambda\}\) is called the exponential family generated by \(Z\). Assuming \(\Lambda\) open, the paper is mainly concerned with the existence of a nondegenerate limit vector \(W\) so that \(\alpha^{-1}_\lambda(Z_\lambda)\Rightarrow W\) as \(\lambda\) tends to the boundary of \(\Lambda\), where \(\alpha_\lambda\), \(\lambda\in \Lambda\), are affine transformations on \(E\). Then the authors are interested in the stability of the exponential family generated by \(W\), and in its domain of attraction. The paper is expository, but the exposition is rather clumsy.