Moment Analysis of Distributions (MAD)

Abstract

The main discussion will be on the well-known fact: Any distribution with finite moments is either M-determinate (uniquely determined by its moments), or M-indeterminate (non-unique). The determinacy properties are important for both, probability theory and statistical inference. Some new aspects will be illustrated. Key words: moment problem; conditions of Cramer, Hardy, Carleman, Hausdorff, Krein, Lin, Pakes;? rate of growth of the moments; Stieltjes classes; stochastic differential equations (SDE); statistical inference problems. New results will be presented: (a) to describe the tails of products of random variables; (b) to use moment type conditions for two distributions to coincide or to be different; (c) to provide explicit constructions of classes of stochastic processes, solutions to SDEs, with preliminary prescribed moment properties. Most of the results have been developed and/or discussed with G.D. Lin. Some results are published, others are in good progress. There are still open questions. Acknowledgment: My first visit to the Institute of Statistical Science, Academia Sinica, Taipei, was in August 1998. This is my visit # 8. Some (perhaps good and useful) work was completed over these years. I am more than grateful to so many colleagues, administration stuff and librarians for their attention, support and good memories.