Reproducibility in the one-parameter exponential family (Q1069580)

Let \(X_ 1,...,X_ n\) be i.i.d. random variables with common distribution an element of a linear one-parameter exponential family \({\mathcal F}=\{F_{\theta}:\theta \in \Theta \subset R\}\) indexed by a natural parameter \(\theta\). It is proved that the distribution of \(\sum^{n}_{i=1}X_ i\) is an element of \({\mathcal F}\), for all \(\theta\in \Theta\) and \(n=1,2,...\), if and only if \({\mathcal F}\) is a family of scale transformed Poisson distributions.