Probability Distributions, Characterization in Terms of Their Moments.
Bulgarian Academy of Sciences, Sofia, Bulgaria, and Shandong University, Jinan, China
Thursday, May 11, 2023
Conference room, Institute of Mathematics
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.