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(July 4th) Characterization of Probability Distributions in Terms of Their Moments


Characterization of Probability Distributions in Terms of Their Moments


2019.07.04 (Thu) 14:00


Jordan Stoyanov (Bulgarian Academy of Sciences)


Wooribyul Seminar Room(E3-2, #2201)


Our discussion will be on distributions, continuous or discrete, with finite all moments of positive integer order. Any such a distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). We are going to explain why some distributions are M-determinate, and others are M-indeterminate. The uniqueness, or M-determinacy, is an important property from both theoretical and applied point of view. In particular, the M-determinacy is an essential requirement for the validity of a fundamental limit theorem. We will describe the current state of arts and concentrate on a variety of checkable conditions which are either sufficient or necessary for uniqueness or for non-uniqueness (Cramer, Carleman, Hardy, Krein, Lin, rate of growth of moments, etc.)

Besides the moments, we will exploit the cumulants/semi-invariants to establish a non-conventional limit theorem. We illustrate ideas and results by examples based on distributions such that Normal, Poisson, Exponential, Lognormal. It time permits, some challenging open questions will be outlined.  


Honorary Professor, Bulgarian Academy of Sciences, Sofia, Bulgaria
Visiting Professor, Shandong University, Jinan, China
Formerly with Newcastle University, UK (1998-2015)
Author of 80 papers and 5 books including Counterexamples in Probability (Third ed.) Dover Publications, New York, December 2013