When Do the Moments Uniquely Identify a Distribution

Abstract

The authors establish when do the moments E(Xh), for h in some subset C of IR, uniquely identify the distribution of any positive random variable X, that is, when is xh a separating function. The simple necessary and sufficient condition is shown to be related with the existence of the moment generating function of the random variable Y = logX. The subset C of IR is thus the set of values of h for which the moment generating function of Y is defined. Examples of random variables characterized in this way by the set of their h-th moments are given.