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量肈Moment Analysis of Distributions: Some Recent Developments

簍量Professor Jordan Stoyanov
School of Mathematics & Statistics, University of Newcastle, UK

丁20057る11ら(琍戳)と10:30-12:00

翴いァ╯皘参璸厩╯┮加ユ剿芔

“穦と1010参璸┮加ユ剿芔


篕 璶

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.