Moment determinacy of distributions: recent results and open questions

The discussion will be on heavy tailed distributions (of random variables or stochastic processes) whose all moments are finite. An important question in the classical moment problem is about the uniqueness. Either such a distribution is uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). One of our goals is to describe the current state of art in this area. After a brief summary of known and widely used classical criteria (Cramer, Carleman, Krein, …), we focus our attention on the following recent developments: how to construct Stieltjes classes consisting of infinitely many distributions all with the same moments; Hardy’s criterion for uniqueness; moment problems for multivariate distributions.
A variety of new results and facts will be reported. Some of them are fresh, hence not so well-known; others look surprising and even shocking. We will illustrate the practical importance of the moment (in)determinacy of distributions in areas such as Box-Cox transformations of random data and Financial modelling, e.g., in option pricing.
Open questions will be discussed during or after the talk. Topics: the non-infinite divisibility of families of M-indeterminate distributions; the maximum error we make when dealing with such distributions; M-(in)determinacy of the distributions of functionals of random sequences and processes.