Asymptotic and Analytic Properties of Mixture Probability Models and Their Application to the Analysis of Complex Systems

The paper presents a survey of the results obtained by the members of the Department of Mathematical Statistics in the field of analytic and asymptotic properties of mixture probability models. Much attention is paid to the representation of some widely applied absolutely continuous probability distributions (gamma, Weibull, Student, Snedecor–Fisher, Mittag-Leffler, Burr, etc.) as mixtures of distributions possessing maximum differential entropy (normal and exponential). Some useful discrete distributions that are representable as mixed Poisson distributions are discussed as well. Examples are presented of limit theorems for statistics constructed from samples with random sizes in which the distributions mentioned above are limit laws. Also, the estimates of convergence rate in these theorems are presented. Some problems related to the application of methods of intellectual analysis of big arrays of dynamically accumulated data based on mixture probability models are discussed.