Moment Analysis of Distributions: Some Recent Developments

Abstract

The main discussion will be on distributions (one- or multi-dimensional) and their characterization properties in terms of the moments: the classical problem of moments. We cover both cases when the solution is M-unique (M- determinate) or M-nonunique (M-indeterminate). Several recent results will be reported. (a) We start with some classical criteria: (i) Why does the Carleman condition imply M-uniqueness? (ii) Can we relax the Carleman condition? (iii) It turns out, a little surprisingly, that M-uniqueness follows even from the (1/2)- Cramer condition! (b) Then we consider functional transformations, called Box-Cox transformations, of random data. The data may come from random variables or stochastic processes. What can we say about the M-determinacy or M- indeterminacy of the distributions involved? One effective approach is: Use the Krein-Lin techniques! (c) How to deal with the non-uniqueness phenomenon? "Not easy", in general, but one possibility is: Construct Stieltjes classes of distributions and calculate their Index of dissimilarity. (d) Why are uniqueness and non-uniqueness important for both practice and theory? We give four illustrations. The first is related to the hot topic of the day: climate changes and global warming (!). The second one concerns properties of distributions such as symmetry, unimodality and infinite divisibility. The third impressive one is from the very popular topic "stochastic financial modelling" (e.g. in Black-Scholes type models): option pricing problem and a related stochastic optimization problem! Finally, the fourth one deals with identifiability of mixtures of distributions. It will be clear that Moment Analysis of Distributions is a rich area of research which is closely related to important practical applications. However, there are challenging open questions. If time permits some of them will be briefly described. The material will be presented in an understandable and (hopefully) attractive way, addressing the talk not only to professionals in the area of statistics, probability, mathematics (theory and/or applications), but also to doctoral and master students.