Moment determinacy of probability distributions

We deal with distributions (or measures), one-dimensional or multi-dimensional, with finite all moments. It is well-known that any such a distribution is either uniquely determined by its moment (M-determinate) or it is non-unique (M-indeterminate). This is the classical moment problem originated in works by Chebyshev, Markov and Stieltjes. Well-known are general conditions which are "iff", but they cannot be checked. Thus our discussion will be on easier and checkable conditions for either uniqueness or non-uniqueness applied to probability distributions. The emphasis will be on some recent developments such as:

• Krein's condition. Converse Krein's condition and Lin's condition.
• Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
• Hardy's condition. Multidimensional moment problem.
• Rate of growth of the moments for (in)determinacy.

There will be results, some of them very new, hints for their proof, examples and counterexamples, and also open questions.