Characterizations of the exponential family by generating functions and recursion relations for moments

Abstract

The exponential family is characterized by the structure of different generating functions and by recursion relations for moments and cumulants. the derived characteriza-tion theorems generalize all previous results on this subject. especially, the assumption of linearity of the exponential family is dropped.

Furthermore, the distribution of a sum of independent exponential family distributed random variables is considered and an equivalent condition is derived for the sum being exponential family distributed.