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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20057る11ら(琍戳)と10:3012:00
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The main discussion will be on distributions (one or multidimensional) and their characterization properties in terms of the moments: the classical problem of moments. We cover both cases when the solution is Munique (Mdeterminate) or Mnonunique (Mindeterminate). Several recent results will be reported.
(a) We start with some classical criteria: (i) Why does the Carleman condition imply Muniqueness? (ii) Can we relax the Carleman condition? (iii) It turns out, a little surprisingly, that Muniqueness follows even from the (1/2)Cramer condition!
(b) Then we consider functional transformations, called BoxCox transformations, of random data. The data may come from random variables or stochastic processes. What can we say about the Mdeterminacy or Mindeterminacy of the distributions involved? One effective approach is: Use the KreinLin techniques!
(c) How to deal with the nonuniqueness 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 nonuniqueness 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 BlackScholes 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.
