REPORTS OF THE SPRING SEMESTER OF 2021
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This is the archive page of the spring semester of 2021 of the United Seminar of the Department of Probability Theory of the Faculty of Mechanics and Mathematics of Moscow State University.
Head of the seminar: academician of the RAS, professor Albert N. Shiryaev
Coordinator of the seminar in spring 2021: professor Elena B. Yarovaya
Secretary of the seminar in spring 2021: Vladimir A. Kutsenko
To subscribe to the seminar newsletter or submit a talk, click "Subscribe or submit" at the top of the page.
February 10, 16:45 MSK
Jordan (Dancho) Stoyanov, Institute of Mathematics & Informatics, Bulgarian Academy of Sciences
Random variables, distributions, functional equations
If X is a random variable with a distribution function F, it is always useful to have properties which characterize uniquely F, hence also X. All started almost a century ago when Polya, Cramer and Raikov established characterizations of the normal and the Poisson distributions. Over the years, characterization problems have been successfully solved by diverse methods. One of them is to use functional equations in terms of Laplace-Stieltjes transforms. In this talk we show that there are classes of functional equations related to nonlinear equations of the type Z = X + TZ, where X, T, Z are nonnegative random variables and “=” means equality in distribution. The problem is to find the distribution function F of X assuming that the distribution of T is known, while the distribution of Z is defined via F. It is important to mention that the uniqueness of the solution of a functional equation (“fixed point”) is equivalent to a characterization property of a distribution. We are going to present new results or extensions of previous results. In particular, we provide another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We give explicit illustrative examples and deal with related topics.
A recording of the report can be found at this link.
The materials of the report can be found at this link.
February 24, 16:45 MSK
Igor Rodionov, Trapeznikov Institute of Control Sciences
Distribution of extremal values of stochastic systems
Let Xn = ( X1,n,…,Xd,n ), d = d(n), n ∈ N – be a sequence of random vectors. Consider the asymptotic distribution of maxima Mn = max1≤i≤d Xi,n, n ∈ N. Clear, if for every x ∈ R it holds
and the asymptotic distribution of maxima of independent copies of Xi,n, is known, then the asymptotic distribution of Mn is known as well. In 1974 Leadbetter proved (1) for Xn – the first n elements of stationary sequence, under the mixing condition D and declustering condition D', and in 1986 a similar result for non-stationary sequence was derived by Hüsler.
We are going to present new sufficient conditions for (1). The proposed approach allows us to generalize Leadbetter's and Hüsler's results and to derive the first general results in this field for random fields in Zk. We will also discuss the applications of this result to Gaussian systems and random graphs, and other author's results in extreme value theory.
A recording of the report can be found at this link.
The materials of the report can be found at this link.
March 03, 16:45 MSK
Vladimir Bogachev, Lomonosov Moscow State University
Uniqueness problems for Fokker – Planck – Kolmogorov equations
This talk concerns several elementary formulated uniqueness problems connected with the stationary Kolmogorov equation and the Cauchy problem for the evolution Fokker – Planck – Kolmogorov equation. Some of these problems, posed by Kolmogorov himself in the 1930s or naturally connected with Kolmogorov's questions, have been solved relatively recently, but others remain open. We shall mainly discuss the case of the unit diffusion coefficient and a smooth drift b, where the stationary equation with respect to a measure µ has the form ∆µ − div(bµ) = 0 or the same form for the solution density, and the parabolic equation for the density p(x,t) is written as ∂t p = ∆p − div(bp). Such equations can be considered in the class of probability measures as well as in the class of bounded (possibly, signed) measures, which leads to different interesting problems. The uniqueness problems we discuss are also related to stationary measures of diffusions, Chapman – Kolmogorov equations and semigroups generated by second order elliptic operators. For understanding the essence of our main problems it is enough to have acquaintance with two semesters of calculus.
A recording of the report can be found at this link.
March 17, 16:45 MSK
Alexander Lykov, Lomonosov Moscow State University
Stability problems in multi-component random systems.
The main topic of our talk is the stability of multi-component random systems with respect to 1) random external influence or 2) random perturbation of the initial data. Hamiltonian systems of point particles with special interaction potential will be used as an essential model of multi-component random systems. At first, we will discuss a stability problem with respect to random external noises: formulate several results about convergence to equilibrium, describe resonances (transience) and conditions for them. Convergence to equilibrium we associate with stability (however, in some sense it quite unjustified). A couple of groups of external noises will be considered: white noise, Gaussian processes, flips, elastic collisions. In the second part of our talk, we will consider a regularity property (non-intersection of trajectories) for multi-component random systems. We will discuss how perturbations (random and deterministic) of the initial data affect this property. We will show some applications of the regularity property to the problem of the derivation of hydrodynamic equations and traffic flow problems.
A recording of the report can be found at this link.
March 31, 16:45 MSK
Evgeny Burnaev, Skoltech, head of ADASE research group
Generative Modeling based on Deep Neural Networks
Predictive Modeling tasks deal with high-dimensional data, and the curse of dimensionality is an obstacle to using many methods for their solutions. In many applications, real-world data occupy only a tiny part of high-dimensional observation space whose intrinsic dimension is essentially lower than the dimension of the space. A popular model for such data is a Manifold model under which data lie on or near an unknaown low-dimensional Data manifold (DM) embedded in an ambient high-dimensional space. Manifold learning is about the predictive modeling of data under this assumption with a general goal to discover a low-dimensional structure of high-dimensional manifold valued data from a given dataset. If dataset points are sampled according to an unknown probability measure on the DM, we face statistical problems about manifold valued data. In the talk, we will discuss some approaches for constructing generative models based on deep neural networks that allow modeling distribution of data "living" on some manifold.
A recording of the report can be found at this link.
April 7, 16:45 MSK
Stanislav Molchanov, The University of North Carolina at Charlotte, Higher School of Economics
Population dynamics in the random environment
The talk will contain the review of the recent results on the existence of the statistical equilibria (steady states) for the models of the population dynamics in the random environment. The environment can be stationary one (time-independent) or non-stationary with the fast oscillation of the parameters of the system in space in time. Technically, the problems of this type are related to the phenomena of localization and intermittency for the random Schroedinger operators or for SPDE’s (stochastic differential equations in the Hilbert space).
A recording of the report can be found at this link.
April 14, 16:45 MSK
Michael Grabchak, Isaac Sonin, The University of North Carolina at Charlotte
A Zero-One Law for Markov Chains (in English)
We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event A from the tail-algebra of MC, for large n, with probability near one, the trajectories of the MC are in states, where P(A|i) is either near 0 or near 1. A similar statement holds for the entrance sigma-algebra, when n tends to minus infinity. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains in both the finite and countable cases. This extends a well-known result due to Kolmogorov. Further, in our discussion, we note an interesting dichotomy between two commonly used definitions of MCs.
A recording of the report can be found at this link.
April 21, 16:45 MSK
Yuliana Linke, Sobolev Institute of Mathematics
Constructing explicit estimators in nonlinear regression with applications to nonparametric regression models.
In nonlinear regression problems, asymptotically optimal estimators are usually given implicitly in the form of solutions of certain equations, while often there are several roots of an equation that determines the estimator of such a kind. The last fact is the main problem, complicating the use of numerical methods. These difficulties can be overcome with the help of one-step estimation procedures, popular in modern Western statistical literature, dating back to the papers by R. Fisher. A one-step estimator is in fact one step of Newton’s method, starting from some initial consistent estimator and asymptotically has the accuracy of the required statistics, which is the root of the corresponding equation. In the talk, firstly, a method will be proposed for constructing explicit consistent with some speed, estimators of parameters for a wide class of nonlinear regression models. In relation to the one-step estimating procedures mentioned above, these new estimators can be used as initial ones. Secondly, we will discuss the asymptotic properties of some types of one-step estimators constructed from nonidentically distributed sample data and which are explicit approximations for consistent M-estimators (for example, quasi-likelihood estimators, least squares ones, etc., in nonlinear regression problems). In addition, explicit estimators of the regression function will be proposed for a wide class of nonparametric regression models, uniformly consistent under mild constraints on the correlation of design elements. Note that in constructing such estimators in nonlinear and nonparametric regression, similar ideas and restrictions on the dependence of design elements are used.
A recording of the report can be found at this link.
April 21, 18:10 MSK
Lomonosov readings
M. Boldin, Lomonosov Moscow State University
On Kolmogorov-Smirnov and Pearson type criteria for checking the normality of autoregression
Autoregressive schemes with a nonzero mean are considered in two situations. In the first one, the autoregressive sequence is observed without errors. The Kolmogorov-Smirnov and Kramer-Mises-Smirnov type criteria are constructed to test the normality of autoregression. The asymptotic distributions of test statistics under the hypothesis and local alternatives are found.
In the second situation, observations may contain errors whose distribution is unknown and arbitrary. A special symmetrized Pearson chi-square test is constructed to test the normality of autoregression. The distribution of test statistics under the hypothesis and local alternatives is found. The qualitative robustness of the test is established.
Ivanov A.О., Malyshev V.A., МГУ им. М.В. Ломоносова
Static Coulomb Clusters
We give a short review of Gibbs distributions for Coulomb point particle system with zero and non-zero temperature. Then the notion of Coulomb cluster is introduced and examples of stable equilibrium for systems of Coulomb clusters are given. We present also a bibliography that shows that the notion of charged clusters is very important in physics, chemistry and biology.
A. Kondratenko, V. Sobolev, Lomonosov Moscow State University
On a property of a convolution with a uniform distribution
Unfortunately, at students seminars on Probability Theory, the topic of convolution of the distributions traditionally is studied one-sidedly. More specifically, the extremely high computational complexity of convolution products leads to the replacement of direct calculations with the application of the central limit theorem. Because of this, many interesting the statement remains unspoken. For example, the convolution modulo N of any discrete integer random variable and a discrete uniform one from 0 to N-1 ( on the numbers 0, 1,..., N-1 ) has a uniform distribution. The report is devoted to the development of this statement for the convolution of arbitrary and absolutely continuous random variables.
April 28, 16:45 MSK
Lomonosov readings
A. Veretennikov, Lomonosov Moscow State University
On estimates of the convergence rate for Markov processes
There is a classical exponential bound for the rate of convergence in the Ergodic theorem for many « simple» (irreducible and acyclic) ergodic homogeneous Markov chains, suggested by A.A.Markov himself in the case of finite state spaces. A.N.Kolmogorov showed that the same bound is applicable in more general situations including a non-homogeneous case. The talk will be devoted to one extension of this bound.
Zamyatin A. A., Malyshev V. A., Lomonosov Moscow State University
Nonequilibrium Boltzmann – convergence to stable flows.
We consider a system of N particles on the circle with mutual interaction and external forces. We discuss the regularity (conservation the order of particles) conditions and convergence (for any initial conditions) to stable particle flow.
Illarionov E. A., Sokolov D. D., Lomonosov Moscow State University
The role of flow anisotropy and finite memory time in estimating the rate of magnetic energy growth in a random flow of a conducting medium
Magnetic fields excited by hydromagnetic dynamo are common in astrophysical bodies and since recently are accessible in laboratory experiments and reproduced in numerical simulations at least for an appropriate range of dynamo governing parameters. However, many realistic dynamo problems, e.g. in the astrophysical framework, propagate in a wider parametric domain where numerical simulations become inaccessible. It remains actual to elaborate an analytical approach at least for qualitative estimation of the processes. The bulk of analytical results accumulated in dynamo studies during a half century of intensive dynamo studies is obviously limited by various simplifications exploited. In particular, simplifications of statistically isotropic and homogeneous random flows and/or short correlation time approximation are usual. In our research we relax some of these restrictions and investigate to what extent it affects the magnetic energy growth (second statistical moment of the magnetic field). We consider 2D and 3D problems and assume finite memory time and axisymmetric correlation tensor for the velocities field (following the Chandrasekhar’s turbulence model). In 2D case we obtain explicit analytical results and in 3D case we obtain a series approximation with respect to small parameters. The proposed method can be naturally extrapolated on the growth rate estimation of higher-order statistical moments.
Kondratenko A. E., Sobolev V. N., Lomonosov Moscow State University On a property of a convolution with a uniform distribution
Many asymptotic expansions in the central limit theorem are obtained using various types of expansions of the characteristic function. One of such types is expansions, in the main part of which there is the last known moment of the original distribution. The idea of constructing such expansions seems to belong to X. Pravits; they were studied in more detail by I.G. Shevtsova.
In 2016 a modification of these expansions for the characteristic function of symmetric distributions was proposed by V.V. Senatov. He considered two cases: in the first his expansion, the fourth moment of the original distribution was present, in the second – the sixth. Their use allowed him to obtain new asymptotic expansions in the central limit theorem for symmetric distributions. This report shows the possibility of constructing such asymptotic expansions in the central limit theorem of any length.
May 12, 16:45 MSK
Elena Bashtova, Lomonosov Moscow State University
Strong Gaussian approximation in queueing theory
The talk will present the results on the strong Gaussian approximation for regenerative flows and the applications of these results in the queueing theory.
The first theorem establishing the strong Gaussian approximation of random processes is due to Strassen (1964). The fundamental achievement was by Komlos, Major and Tusnady (1975-1976), who obtained accurate estimates in the strong Gaussian approximation of the sums of independent identically distributed random variables. This result initiated a whole direction of research aimed at obtaining estimates (as accurate as possible) for multidimensional systems, weakly dependent sequences, random fields, etc.
It should be noted that sharp estimation of the rate of convergence is a nontrivial problem. It is enough to say that the exponential rate of convergence for sums of bounded functionals of geometrically ergodic Markov chains, was established only in 2015 by Merleved and Rio via a sophisticated coupling technique. At the same time, a straightforward combination of results for sums of i.i.d.r.v. and the Skorokhod embedding method gives only the rate t-1/4.
We consider processes which are a generalization of many types of processes used in queueing and insurance theory. In particular, for the sums of functionals of Markov chains it is the case. Further, many processes in queueing theory are Lipschitz functionals in the trajectories of the incoming flows and the service processes. Therefore theorems on the strong Gaussian approximation for regenerative flows make way to construct a strong approximation for such processes as the queue length or the waiting time in heavily loaded systems. Several examples of such approximation will be presented in the talk.
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