Reproducibility and natural exponential families with power variance functions (Q1110939)

Let \(X_ 1\),..., \(X_ n\) be independent identically distributed random variables whose common distribution belongs to a family \({\mathcal F}=\{F_{\theta}\in \Theta \subset {\mathbb{R}}\}\) indexed by a parameter \(\theta\). We say that \({\mathcal F}\) is reproducible if there exists a sequence \(\{\) \(\alpha\) (n)\(\}\) such that \[ {\mathcal L}(\alpha (n)\sum^{n}_{i=1}X_ i)\in {\mathcal F}\quad for\quad all\quad \theta \in \Theta \quad and\quad n=1,2,.... \] This property is investigated in connection with linear exponential families of order 1 and its intimate relationship to such families having a power variance function is demonstrated. Moreover, the role of such families is examined, via a unified approach, with respect to properties related to infinite divisibility, steepness, convolution, stability, self-decomposability, unimodality, and cumulants.