Reproducibility and Natural Exponential Families with Power Variance Functions

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. r.v.'s having common distribution belonging to a family $\mathscr{F} = \{F_\theta: \theta \in \Theta \subset R\}$ indexed by a parameter $\theta$. $\mathscr{F}$ is said to be reproducible if there exists a sequence $\{\alpha(n)\}$ such that $\mathscr{L}(\alpha(n)\sum^n_{i=1} X_i) \in \mathscr{F}$ for all $\theta \in \Theta$ and $n = 1, 2, \cdots$. 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, in a unified approach, with respect to properties relative to infinite divisibility, steepness, convolution, stability, self-decomposability, unimodality, and cumulants.