A moment problem for random discrete measures Kondratiev, Yuri Kondratiev Yuri Kuna, Tobias Kuna Tobias Lytvynov, Eugene Lytvynov Eugene Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures n on X which are of the form eta = Sigma(i) s(i)delta(xi), where, for each i, s(i) > 0 and delta(xi) is the Dirac measure at x(i) is an element of X. A random discrete measure on X is a probability measure on 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 kt is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure mu. 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. (C) 2015 Elsevier B.V. All rights reserved. 125 9 3541-3569 3541-3569 Elsevier 2015