UNITED SEMINAR OF THE DEPARTMENT OF PROBABILITY THEORY OF LOMONOSOV MOSCOW STATE UNIVERSITY
UNITED SEMINAR OF THE DEPARTMENT OF PROBABILITY THEORY OF LOMONOSOV MOSCOW STATE UNIVERSITY
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.
The seminar is held online every Wednesday from 16:45 to 17:45 Moscow time.
Permanent link to the Zoom room: http://bit.ly/3HY8K6d
Room ID: 844 6792 3144 Access code: 697663
Head of the seminar: academician of the RAS, professor Albert N. Shiryaev
Coordinator of the seminar in autumn 2024: professor Elena B. Yarovaya
Secretary of the seminar in autumn 2024: Oleg E. Ivlev
To subscribe to the seminar newsletter or submit a talk, click "Subscribe or submit" at the top of the page.
December 18, 16:45 msk
Iuliia Makarova, Lomonosov MSU, Russia
Branching random walks with a few types of particles on multidimensional lattices
We consider different models of branching random walks (BRW) on multidimensional lattice with continuous time. The first model - BRW with one type of particles in which at every lattice point particles can appear from outside. We get differential equations and asymptotic behaviour for the moments of particles number. The second model - multiple BRW. The main results are dedicated to the moments of subpopulations of particles of each type.
The recording: YouTube
December 11, 16:45 msk
Hanchao Wang, Shandong University, China
The Majorizing Measure Theorems for Log-concave-tailed Canonical Process
The Majorizing measure theorem for Gaussian processes is famous. This talk will introduce my work on majorizing measure theorem for log-concave tailed canonical processes. We fully characterize the expected suprema of log-concave-tailed canonical processes. In particular, we obtain a new majorizing measure theorem and its dual version.
The recording: YouTube
December 4, 16:45 msk
Poisson approximation in terms of the point distance
We present estimates of the accuracy of Poisson approximation to the distribution of a sum of integer-valued random variables in terms of the point distance.
The recording: YouTube
November 27, 16:45 msk
Queueing Networks with Node Dynamics: A Survey of Models, New Results, and Open Problems
This talk addresses queueing networks with a large number of nodes, where node dynamics (e.g., failures and recoveries) lead to changes in network topology. Special attention is devoted to sparse networks, which most accurately represent real-world systems. The presentation will begin with a survey of the most prominent approaches in queueing theory, discussing their advantages and limitations. Subsequently, more general models overcoming the constraints of standard methods will be proposed, accompanied by an analysis of their properties. The talk will conclude with the results of simulation studies, a discussion of open problems, and an outline of key challenges relevant to practical applications.
The recording: YouTube
November 20, 16:45 msk
Large lower local deviations of branching processes in a random environment
We consider large deviation probabilities for a branching process Z_n = X_{n,1} + ... + X_{n,Z_{n-1}} in an i.i.d. random environment. We assume that X_{i,j} for a fixed environment have a geometric distribution and assume that a step of the associated random walk satisfies the Cramer condition. For a supercritical process we study lower large deviation probabilities in a local form. For all types of processes the upper large deviation probabilities in a local form are described. The most interesting result is obtained in the second zone of lower large deviations - we show that probabilities of all values from 1 to some exponentially large level have the same asymptotics.
The recording: YouTube
November 13, 16:45 msk
Spectral theory of the Schrödinger type operators on the Exner’s periodic quantum graphs
Quantum (or metric) graphs are a natural class of physical systems intermediate between lattices Z^d with the lattice Hamiltonians and the continuous models with the phase space R^d and classical Schrödinger operators H = -Δ + σV(x) on L^2(R^d). In the simplest form such a graph Γ^d is a cubic lattice Z^d with 1D edges connecting the neighboring vertices. The edges are equipped by the Euclidean metric, they are connected by the Kirchhoff’s gluing conditions at the vertices. P. Exner introduced this kind of graphs and operators in 60th before the development of the general theory. This topic is popular today mainly because of applications to the optical computers. One of the most important properties of corresponding periodic Hamiltonians is the existence of the infinitely many spectral gaps. It means that such graphs demonstrate (in contrast with the classical theory in R^d, d≥2) the semi-conductor properties for arbitrary high frequencies. The talk will contain the review of recent results on the structure of the Brownian motion on Γ^d and the spectra of the Schrödinger type operators with the decreasing or increasing to ∞ potentials. The most interesting (probably) are the theorems about point spectrum (localization) inside the gaps of the periodic background under random perturbations.
The recording: YouTube
November 6, 16:45 msk
Ekaterina Palamarchuk, CEMI RAS, NRU HSE, Russia
On non-ergodic optimality criteria for time-varying linear stochastic control systems
The talk is devoted to the study of time-varying linear stochastic control systems. In order to derive an optimal control law over an infinite time horizon, non-ergodic optimality criteria are introduced, extending the notions of long-run averages. We consider two special examples of linear time-varying control systems. A system with a time-varying diffusion matrix and a system involving discounting in the cost are examined. We also discuss issues related to convergence rates for non-ergodic criteria.
The recording: YouTube
October 30, 16:45 msk
Ciprian Tudor, University of Lille, France
Multidimensional Stein-Malliavin calculus for Gaussian and Gamma distribution
We develop an extension of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the probability distributions of (X,Y) and (Z,Y), where X,Y are arbitrary random vectors and Z ~ N(0, σ^2) (or it is Gamma distributed) is independent of Y. We apply this method to study the asymptotic independence for sequences of random vectors.
The recording: YouTube
October 23, 16:45 msk
Temirlan Abildaev, St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg University, Russia
Limit theorems associated with the generator of a symmetric Lévy process with the delta potential
We will consider a one-dimensional symmetric Lévy process that has local time. In the first part of the talk, we will construct a self-adjoint extension for the generator of the process so that this extension corresponds to the generator with the delta potential. Using the extension, we will prove the Feyman-Kac formula in the case of the delta function-type potentials and prove a limit theorem for an operator semi-group corresponding to this formula. In the second part, we will construct a one-parameter family of distributions that exponentially attract sample paths of the process to a given point. We will prove a limit theorem for the distribution of a point where the attracted sample path comes and show that the finite-dimensional distributions of the resulting process converge in total variance to the corresponding finite-dimensional distributions of a Feller process.
The recording: YouTube
October 16, 16:45 msk
Jordan Stoyanov, Bulgarian Academy of Sciences, Sofia, Gwo Dong Lin, Academia Sinica, Taipei
Normal distribution: recent not so well-known results and some open questions
The main discussion in this talk will be on the following distributions: normal, half-normal, log-normal, inverse Gaussian, skew-normal, log-skew-normal, exp-normal. Of interest to us, and in general, are properties such as characterizations, moment-determinacy, infinite divisibility, comparison. We are going to report a series of well-selected results obtained by ourselves or by other authors over a couple of decades. The results are interesting, some are known, others in the best case are only 'slightly known', but definitely 'not well-known'. All results, statements, corollaries, etc., will be formulated completely and clearly, useful hints will be given, and in two cases, full proofs provided. We hope that such results and the methods used would be useful for university students and teachers. Professional researchers may challenge seven clearly formulated open questions.
The recording: YouTube
October 9, 16:45 msk
Qi-Man Shao, Southern University of Science and Technology, China
Perspective of Self-normalized Limit Theory
Limit theory plays an important role in probability and statistics. Classical limit theorems such as the law of large numbers , the central limit theorem and the Craḿer moderate deviation theorem, under deterministic standardization, have been well developed and understood. However, standardized coefficients in applications are more often random, or self-normalized. In this talk, we shall review recent developments of limit theory for self-normalized processes as well as applications to statistical inference.
The recording: YouTube
October 2, 16:45 msk
Yaroslav Sergeyev, University of Calabria, Rende, Italy and Lobachevsky State University, Nizhni Novgorod, Russia
Computations with numerical infinities and infinitesimals and infinitesimal probabilities
In Kolmogorov’s probability axioms, the axiom of additivity is formulated using notions of countable sets and sigma-algebras. In models having an infinite number of events, e.g., in continuous models, this can lead to some ungracious situations where possible events corresponding to sets of measure zero have probability zero whereas in the discrete finite case the probability zero is assigned to the impossible event only.
In this lecture, a recent award-winning computational methodology (not related to non-standard analysis) is described (see [1-4]). It allows people to work with infinities and infinitesimals in a unique framework and in all the situations requiring these notions not only symbolically, but also numerically on a computer. The new methodology is based on the Euclid’s Common Notion “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).
It is shown that the methodology allows one to create a new viewpoint on probability and to avoid a number of paradoxes (see [3-7]) related to infinity, infinitesimals, and probability (among other paradoxes that can be avoided we find classical paradoxes of Hilbert, Galilei, Torricelli, etc.). One of the strong advantages of this methodology consists of its usefulness in practical applications (see [1, 4, 8, 9]) that can be implemented on a new kind of supercomputer called the Infinity Computer patented in several countries. It works numerically with numbers that can have several infinite and infinitesimal parts using a special floating-point representation. It should be also stressed that the methodology is already successfully taught in colleges in several countries (see, e.g., [7, 10] and references given therein). We conclude by emphasizing that reviews, videos, and more than 70 papers of authors using this methodology in their research areas can be downloaded from https://www.theinfinitycomputer.com
Selected references
1. Sergeyev Ya.D. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 4(2), 219–320, 2017.
2. Sergeyev Ya.D., Arithmetic of Infinity, Edizioni Orizzonti Meridionali, CS, 2003 (2nd ed. 2013).
3. Sergeyev Ya.D. Some paradoxes of infinity revisited, Mediterr. Journal of Mathem., 19, article 143, 2022.
4. Sergeyev Ya. D. A new look at infinitely large and infinitely small quantities: Methodological foundations and practical calculations with these numbers on a computer. Informatics and Education, 36(8), 5–22, 2021. (In Russian.)
5. Calude C. S., Dumitrescu M. Infinitesimal probabilities based on Grossone, SN Comput. Sci., 1, 36, 2020.
6. Rizza D. A Study of Mathematical Determination through Bertrand’s Paradox, Philosophia Mathematica, 26(3), 375–395, 2018.
7. Nasr L. Students’ resolutions of some paradoxes of infinity in the lens of the Grossone methodology. Informatics and Education, 38(1), 83–91. 2023.
8. Sergeyev Ya.D., De Leone R. (eds.) Numerical Infinities and Infinitesimals in Optimization, Springer, Cham, 2022.
9. Falcone A., Garro A., Mukhametzhanov M.S., Sergeyev Ya.D. Advantages of the usage of the Infinity Computer for reducing the Zeno behavior in hybrid system models, Soft Computing, 27(12), 8189-8208, 2023.
10. Rizza D., Primi Passi nell’Aritmetica dell’Infinito, Bonomo Editore, Bologna, 2023.
The recording: YouTube
September 25, 16:45 msk
Shige Peng, Shandong University, China
Theorems, Method and Content of BSDE and Nonlinear Expectation Theory
In this talk, we present our research explorations of backward stochastic differential equations and nonlinear expectations. We first recall the basic definition of a space of linear, as well as nonlinear expectations, and then, through the representation theorem and examples of nonlinear i.i.d. (independent and identically distributed), and explain why this new framework can be applied to it’s own stochastic calculus and quantitatively analyze probabilistic and distributional dynamical uncertainty hidden behind data sequence.
We have introduced two fundamentally important nonlinear Gaussian distribution and maxima distribution, then corresponding nonlinear law of large numbers (LLN) and nonlinear central limit theorem (CLT), which are crucial and fundamental breakthroughs. A typical application is a basic max-mean algorithm.
We also present a basic continuous-time stochastic process, which is nonlinear Brownian motion (G-BM) and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has been deeply and strongly influenced by the axiomatic probability theory founded by Kolmogorov (1933). The key idea is to directly introduce the fundamental notion of nonlinear expectation Ê. The special case of the linear expectation corresponds a probability space (Ω, F, P). It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world data.
The recording: YouTube