Moment Analysis of
Probability Distributions
报告专家：Prof. Jordan Stoyanov（Bulgarian Academy of Sciences & Shandong University）
报告时间：
2025年6月30日（星期一）下午3:00-5:00
2025年7月2日（星期三）下午3:00-5:00
2025年7月4日（星期五）下午3:00-5:00
课程地点：数学学院西202
课程摘要：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:
List of key words and phrases: Stieltjes, Hamburger and Hausdorff moment problems; non-uniqueness phenomenon; conditions for uniqueness; conditions for non-uniqueness: Cramer, Hardy, Carleman, Krein; Stieltjes classes for M-indet distributions; index of dissimilarity; rate of growth of the moments; Box-Cox transformations of random data; Frechet-Shohat Theorem; random sums and random products; symmetry, unimodality and infinite divisibility; moment problem for multivariate distributions.
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’.
专家简介：Jordan Stoyanov (JS for short) is Honorary Professor of Bulgarian Academy of Sciences and Visiting Prof. at Shandong University. Graduated from Moscow State University earning his scientific degrees in Probability Theory under the supervision of Academician Albert Shiryaev of the Kolmogorov’s School in Probability. JS worded in Bulgaria, and was visiting professor at many universities all over the world. JS has published more than 80 papers and 5 books. Well-known is his book “Counterexamples in Probability”, three editions in English and two in Russian.
邀请人：胡泽春
