Probability Distributions, Characterization in Terms of Their Moments.

We deal with distributions, continuous or discrete, with finite all moments of positive integer orders. Any such a distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). It will become clear why some distributions are M-determinate, and others are M-indeterminate and what happens under nonlinear (Box-Cox) transformations of random data.

There are classical and well-known results and conditions of the type “iff” for M-determinacy, expressed in terms of the smallest eigenvalues of Hankel matrices. They are in the category `uncheckable’. The main attention will be paid to `checkable’ conditions, either sufficient or necessary for uniqueness or for non-uniqueness (Cramer, Carleman, Hardy, Krein, Lin, rate of growth of moments, etc). Some recent results will be reported, e.g․, a nonconventional limit theorem, based on the cumulants. We will provide illustrations by distributions such as Normal, Log-normal, Poisson, Exponential. Challenging open questions will be outlined. Some of the results are joint with G.D. Lin, P. Kopanov, C. Vignat.