Reproducibility and natural exponential families with power variance functions

Let X1,⋯,Xn be i.i.d. r.v.'s having common distribution belonging to a family F={Fθ:θ∈Θ⊂R} indexed by a parameter θ. F is said to be reproducible if there exists a sequence {α(n)} such that L(α(n)∑ni=1Xi)∈F for all θ∈Θ and 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, in a unified approach, with respect to properties relative to infinite divisibility, steepness, convolution, stability, self-decomposability, unimodality, and cumulants.