Moment Analysis of Probability Distributions

The main discussion will be on probability/statistical distributions, discrete or continuous, one-dimensional or multidimensional. One of our goals is to use the moments of integer positive order, also the related cumulants, and analyse their role in deriving important distributional properties. With all moments finite, we have a dichotomy, a distribution is either M-determinate, M-det, = uniquely determined, or it is non-unique, M-indeterminate, M-indet. The M-det is related to several useful properties, the M-indet is quite ‘risky’.

Available in the literature are variety of different conditions for M-det or M-indet. There are uncheckable conditions, practically of no use. The main attention will be on checkable conditions, sufficient or necessary, for either M-det or M-indet property. Two questions will be considered in detail: characterizations and limit theorems in terms of the moments or the cumulants. Specific items, not all, will be chosen from the following: After briefly covering classical and well-known results, the emphasis will be on new and recent developments based on diverse ideas and techniques. The results will be illustrated by distributions such as N, LogN, Exp, Gamma, Poisson, LogPoisson, etc.

The course will be addressed mainly to MSc and PhD students specializing in Statistics, Probability and Analysis, also to young researchers in these areas. Professionals may find challenging some explicitly outlined open questions.

A useful source is the recent paper, coming soon in Springer, see there some 69 references: J. Stoyanov, “Normal Distribution: Some Recent Results and Twelve Open Questions”.