A moment problem for random discrete measures

Let XX be a locally compact Polish space. A random measure on XX is a probability measure on the space of all (nonnegative) Radon measures on XX. Denote by K(X)K(X) the cone of all Radon measures ηη on XX which are of the form η=∑isiδxiη=∑isiδxi, where, for each ii, si>0si>0 and δxiδxi is the Dirac measure at xi∈Xxi∈X. A random discrete measure on XX is a probability measure on K(X)K(X). The main result of the paper states a necessary and sufficient condition (conditional upon a mild a priori   bound) when a random measure μμ is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure μμ. Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterization via moments is given when a random measure is a point process.