Inference Problems Involving Moment Determinacy of Distributions

We describe diverse stochastic inference problems whose solution essentially depends on the moment determinacy of some distributions involved. For a variety of stochastic models we ask questions such as “how to identify a distribution if knowing its moments?” “how asymmetric can be a distribution with zero odd order moments?” “is any mixture model identifiable?” For specific models we provide answers, motivating arguments, and illustrations. Some challenging open questions are outlined.