Moment determinacy of distributions

The main discussion will be on distributions with finite all moments. Any such a distribution is either M-determinate (unique by the moments) or M-indeterminate (non-unique). Along with classical criteria, some recent and interesting developments will be reported. E.g., we can analyze the moment determinacy of Box-Cox transformations of random data. Results concerning stochastic processes will also be presented. The illustrative examples will involve popular distributions such as normal, log normal, exponential, Poisson, … Some not so-well known and even shocking facts will be discussed. It will be clear that the moment determinacy and/or indeterminacy of distributions are essential in statistical inference problems. If time permits, open questions will be outlined.