This is the page of the United Seminar of the Department of Probability Theory of the Faculty of Mechanics and Mathematics of Moscow State University. The permanent website of the seminar is here. The seminar is a continuation of the research seminar of the Department of Probability Theory under the leadership of A.N. Kolmogorov and B.V. Gnedenko.

Upcoming reports

Past reports

February 21, 16:45 MSK
Evgeny Burnaev, Skoltech, AIRI

From stochastic differential equations to the Monge-Kantorovich problem and back: the path to artificial intelligence?

A.N. Kolmogorov is the greatest mathematician of the XX century, the founder of modern probability theory, who also laid the foundations of the theory of Markov random processes in continuous time. These results, which significantly impacted the development of applied methods of signal processing, filtering, modeling, and processing of financial data, were again in the spotlight due to the development of artificial intelligence and its applications in the 21st century. Indeed, to solve such important applied tasks as image super-resolution, text-to-speech synthesis, image generation based on text descriptions, etc., effective generative modeling methods are required to generate objects from the distribution represented by a sample of examples. Recent achievements in the field of generative modeling are based on diffusion models and use the mathematical foundations laid down in the last century by A.N. Kolmogorov and his followers. I will talk about modern approaches to generative modeling based on the diffusion processes and on the solution to the Monge-Kantorovich problem. I will show the connection between the entropy-regularized Monge-Kantorovich problem and the problem of constructing a diffusion process with specific extreme properties. I will demonstrate applications of the corresponding algorithms based on various image processing problems.

March 06, 16:45 MSK
Zhonggen Su, School of Mathematical Sciences, Zhejiang University, Hangzhou, China

Three-parameter distributional approximations for sums of locally dependent random variables

Consider a finite family of locally dependent non-negative integer-valued random variables with finite third order moments, and denote by W their sum. There have been a number of research works on computing the distributions of W in literature. In this talk I shall report a recent work on  three-Parameter distributional approximation  for W. Specifically speaking, denote by M a three parameter random variable, say the mixture of Bernoulli binomial distribution and Poisson distribution, the mixture of negative binomial distribution and Poisson distribution or the mixture of Poisson distributions. We use Stein's method to establish general upper error bounds for the total variation distance between W and M, where three parameters in M are uniquely determined by the first three moments of W. As a direct consequence, we obtain a  discretized normal approximation for W. To illustrate, we study in detail a few of well-known examples, among which are counting vertices of all edges point inward, birthday problem, counting monochromatic edges in uniformly colored graphs, and triangles in the Erdős–Rényi random graph. Through delicate analysis and computations, we obtain sharper upper error bounds than existing results. This talk is based on recent joint works with Xiaolin Wang.

The recording: https://youtu.be/ZUC0rFLHrL0 

March 20, 16:45 MSK
Leonid Koralov, University of Maryland, College Park
Stability and Metastability for Degenerate Diffusions

We study diffusion processes in R^d that degenerate at a finite collection of surfaces (or points) in R^d and small perturbations of such processes. Assuming certain ergodic properties at and near the invariant surfaces, we describe the rate at which the process gets attracted to or repelled from the surface, based on the local behavior of the coefficients. For processes that include, additionally, a small non-degenerate perturbation, we describe the meta-stable behavior. Namely, by allowing the time scale to depend on the size of the perturbation, we observe different asymptotic distributions of the process at different time scales. The talk is based on joint work with M. Freidlin. 

April 3, 16:45 MSK

Lomonosov readings

Anatoly  Manita, MSU
Larisa Manita, HSE, MIEM

On multi-dimensional Markov processes inspired by  special randomizations and their applications to modeling of opinion dynamics

We show that mathematical models for some concrete applied problems (such as consensus in social groups, agreement algorithms etc.)  lead to stochastic processes of special type. We discuss how these processes are related to some classical studies in  probability theory, among which the famous results of A.N.Kolmogorov on nonhomogeneous Markov chains. Our study is focused on properties of invariant distributions especially asymptotic when the number of paritipant goes to infinity.

Aleksandr Veretennikov, MSU
Alina Akhmiarova, MSU

On the Law of Large Numbers

New versions of the weak LLN for non-identically distributed random variables possessing a property of weak dependence are established. Under the condition of finite expectations it is always assumed that EX_k = 0. The additional condition is uniform integrability in the sense of Cesàro. The main motivation was to understand what sufficient conditions of weak dependence are required.

April 17, 16:45 MSK
Stanislav Molchanov, UNC Charlotte
Stochastic models of the primes

The idea that the prime numbers are ''random'' in explicit form was formulated by Legendre and Gauss but only Cramer (1936) published the paper where primes considered as the element of the space of the special random sequences. Namely: Cramer called each integer n > 2 ''quasiprime'' with probability  1/ln(n) independently of other numbers n' ≠ n. Put in Bernoulli ansamble N(x,ω) = #{quasiprimes n : n ≤ x}. One of the most important Cramer’s results state that for x → ∞ P-a.s. N(x,ω) = Li(x) + O(√x), Li(x) is offset logarithmic integral from 3 to x.  It is well known fact that the similar estimate N(x) = Li(x)+O(√x)  for the counting function of primes is equivalent to the famous conjecture on the zeros of Riemann’s ζ-function. 

The talk will contain results on the Cramer’s model and it generalizations, on the corresponding ζ-function and results on the numerical experiments with primes on supercomputer. My collaborators in this area are V. Margarint (UNCC), Ya. Malinovsky (Univ. Maryland).