Upper and lower bounds are discussed for the expectations of functionals (from a fairly wide class) of sums of independent identically distributed random variables with values ​​in a linear space. These estimates will be written in terms of the corresponding moments of the accompanying infinitely divisible laws (compound Poisson distributions). This topic goes back to the work by Yu.V. Prokhorov (1962) and is related to the problem of A.N. Kolmogorov regarding the approximation for the distributions of sums of independent random variables by infinitely divisible laws, not necessarily Gaussian or Poisson. As an application, we derrive Poissonization inequalities for one class of additive functionals of empirical point processes.

The talk will focus on the consistency of Nadaraya--Watson estimates in nonparametric regression without using traditional dependency conditions of design elements (regressors). The design can be both fixed and not necessarily regular, and random, while not necessarily satisfying the traditional conditions for the correlation of the design elements. A new characteristic of the dependence of design elements is proposed, in terms of which sufficient conditions are given for the uniform consistency of the Nadaraya--Watson estimates.

The talk considers a one-dimensional random walk model based on the phenomenology of a memory stream. The jumps of this random walks have the structure of a moving average built on a finite sequence of memory functions and some stationary sequence. A physical interpretation of these memory functions and stationary sequence is given. For the normalized random walk under consideration, a limit theorem is obtained in the metric space D[0,1].

The talk will review the results for additive functionals of order statistics. Functionals such as L-statistics, Lorentz curves and L-processes, which are widely used in econometrics, risk insurance, actuarial mathematics, etc., will be considered.

The talk will discuss the results of numerical modeling and analysis of epidemiological data using various parametric and nonparametric regression methods. The main attention will be paid to new classes of locally constant and locally linear kernel estimates in nonparametric regression. A comparison will be made of the current most common statistical methods of estimation and the procedures listed above, discussed in the talk by Yu.Yu. Linke (01.03.22). The results are part of a collaborative study carried out by the group specialists from Sobolev Institute of Mathematics SB RAS (Yu.Yu. Linke, I.S. Borisov, and P.S. Ruzankin), Moscow State University (V.A. Kutsenko and E.B. Yarovaya) and National Research Center for Therapy and Preventive Medicine (S.A. Shalnova).

Weibull-type exponential estimates are obtained for the tails of distributions of infinite polynomial forms based on sequences of dependent random variables. The exponent in the resulting inequalities has the unimprovable order. Applications of these results are considered to the analysis of distributions of multiple stochastic integrals as well as U- and V-statistics of arbitrary order.

In the talk we will discuss two classes of universal kernel-type estimators in nonparametric regression uniformly consistent under close to minimal and illustrative conditions on design points. The universality of these estimators lies in the fact that their asymptotic properties do not depend on the structure of dependence of the design elements, with respect to which the domain of the regression function is supposed to be densely filled in some sense. Some of the results presented in the talk are joint studies with I.S. Borisov, P.S. Ruzankin, E.B. Yarovaya (MSU), V.A. Kutsenko (MSU), and S.A. Shalnova (National Medical Research Center for Therapy and Preventive Medicine).

Exponential upper bounds are obtained for the tails of distributions of centered and normalized number of cycles of fixed length in Erdős-Rényi graphs. The estimates are uniform with respect to the number of vertices in the graph.

In the study of heterogeneous samples taken from finite mixtures of probability distributions, the number of mixing distributions (components) as well as their corresponding weights and parameters, may be unknown. The problem of separation (decomposition) of mixtures is the problem of estimating unknown parameters miscible distributions and their weights. The talk will consider some well-known methods for solving this problem, their advantages and disadvantages. Using these methods, we decompose mixtures of some real socio-economic heterogeneous data, that, in particular, solve the problem of typology (classification) of data, i.e., the selection of groups of observations taken from the same component of the mixture.

Expectation Maximization algorithms for Poisson data will be reviewed.

The achievements and unsolved problems of modern diagnostic nuclear medicine based on the methods of single-photon emission computed tomography (SPECT) and positron emission tomography (PET) will be reviewed. The capabilities of mathematical modeling of a patient examination by the SPECT method making use of the "virtual patient", the "virtual tomograph" and applying statistical techniques to solve the inverse problem of image reconstruction will be discussed. The promising directions of the nuclear medicine of the nearest future, including the development of theranostics and radiomics, will be discussed.