Jordan Stoyanov (Newcastle/Ljubljana)

Probability Distributions and Their Moment Determinacy

Abstract: We deal with probability distributions, one-dimensional or multi-dimensional, whose all positive integer order moments are finite. Either such a distribution is uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). Thus the topic of the talk is in the classical moment problem. Besides recalling briefly the well-known conditions by Cramer, Carleman and Krein, the emphasis will be on some new developments obtained over the last years. Some of the following specific topics will be presented in more details:
(a) New Hardy’s criterion for uniqueness.
(b) Criteria based on the rate of growth of moments.
(c) Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
(d) Multidimensional moment problem.
(e) Nonlinear transformations of random data and their moment (in)determinacy.
(f) M-determinacy of distributions of stochastic processes defined by SDEs.
There will be new and well-referenced results, hints for their proof, and illustrations by examples and counterexamples. Some facts are not so well-known and even look shocking. Intriguing open questions and conjectures will be outlined.