UNITED SEMINAR OF THE DEPARTMENT OF PROBABILITY THEORY OF LOMONOSOV MOSCOW STATE UNIVERSITY
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This is the page of the fall semester of 2022 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/bks_terver_2022
Room ID: 899 0979 6692 Access code: 161752
Head of the seminar: academician of the RAS, professor Albert N. Shiryaev
Coordinator of the seminar in fall 2022: professor Elena B. Yarovaya
Secretary of the seminar in fall 2022: Vladimir A. Kutsenko
To subscribe to the seminar newsletter or submit a talk, click "Subscribe or submit" at the top of the page.
The list of reports from 2020 and previous years can be viewed at this link.
The list of reports from 2021 and 2022 can be in the corresponding sections of this page.
The list of reports from 2020 and previous years can be viewed at this link.
The list of reports from 2021 and 2022 can be in the corresponding sections of this page.
Past reports
Past reports
September 14, 16:45 MSK
Aleksandr Bulinskii, Lomonosov Moscow State University
The Stein method and its development
In 1972 C.Stein proposed new method to prove the central limit theorem. More precisely, a differential equation of the first order was introduced, which is nowadays called the Stein equation. A solution of the mentioned equation along with application of probability metrics permits to estimate the proximity of a distribution under consideration (e.g. the distribution of appropriately normalized partial sums of random variables) to a normal law. Further this powerful method was developed and modified for those situations when the "target distribution'' for comparing with initial one is non-Gaussian. The essential role here is played by various transformations of a probability law (in particular, the zero-biased and equilibrium transformations). In certain cases such techniques combined with other devices gives the sharp estimates. We discuss the problem which basic concepts and results concerning the Stein method should be included in the general course of Probability Theory for students of the Faculty of Mathematics and Mechanics of the Lomonosov Moscow State University. Also we provide the comparison of the considered method with classical ones: moments (semiinvariants) method, characteristic functions, the Lindeberg method, the Trotter approach. The version of the Stein method involving a generator of specified Markov process deserves special attention. The author results and the results of his pupils obtained in the framework of Stein's method are mentioned as well. To complete the picture we tackle the connection of the Stein techniques with the Mallavin calculus.
The recording of the seminar is available here.
September 21, 16:45 MSK
Aleksandr Shklyaev, Lomonosov Moscow State University, Steklov Mathematical Institute of RAS
Large Deviations of Branching Processes in Random Environment with some Generalizations
The abstract is attached below.
The recording of the report is available here.
A.V. Shklyaev.pdfOctober 12, 16:45 MSK
Alexandr Bufetov, HSE, Steklov Mathematical Institute of RAS
The gaussian multiplicative chaos for the sine-process
To almost every realization of the sine-process one naturally assigns a random entire function, the analogue of the Euler product for the sine, the scaling limit of ratios of characteristic polynomials of a random matrix. The main result of the talk is that the square of the absolute value of our random entire function converges to the Gaussian multiplicative chaos.
As a corollary, one obtains that almost every realization with one particle removed is a complete and minimal set for the Paley-Wiener space, whereas if two particles are removed, then the resulting set is a zero set for the Paley-Wiener space. Quasi-invariance of the sine-process under compactly supported diffeomorphisms of the line plays a key role.
The recording of the report is available here.
October 19, 16:45 MSK
Stanislav Molchanov, UNC Charlotte
What may be called the Dickman distribution?
The Dickman distribution had been introduced (at the physical level of rigor) by a German actuary Dickman in the connection with the statistics of the prime factors of the natural numbers. In the last several years there appeared many publications (in the applied probability literature) where other objects are studied and these objects are called the “Dickman distribution”. The review of this huge volume of literature can be found in the recent paper by S.A. Molchanov and V.A. Panov (Russ. Math. Survey, 2020, 75(6)). In my talk I will discuss some properties of these quasi- Dickman distributions, in particular their infinite divisibility and their applications in the graph theory, financial mathematics etc.
The recording of the report is available here.
October 26, 16:45 MSK
Natalia Smorodina, Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
On a limit theorem for branching random walk
The talk is based on the joint work with E.B.Yarovaya (MSU). We consider a continuous-time branching random walk on a multidimensional lattice. We suppose that the transport of particles on the lattice be given by a symmetric, homogeneous and irreducible random walk where the branching intensity at a a point x tends to zero as ||x|| → ∞. Moreover, an additional condition is satisfied for the parameters of the branching random walk, which guarantees exponential growth in time of the mean number of particles at each point of the lattice. Under these assumptions, we prove the limit theorem on the mean square convergence of the normalized number of particles at an arbitrary fixed point of the lattice for t → ∞. The proof is based on the approximation of the normalized number of particles by a nonnegative martingale.
The recording of the report is available here.
November 2, 16:45 MSK
Aleksandr Tikhomirov, Komi Scientific Center of the Ural Branch of the Russian Academy of Sciences
Limit theorems for Laplace matrices of generalized random graphs
We consider the asymptotic behavior of the empirical spectral distribution function of the Laplace matrix of a weighted random graph with an increasing number of vertices. Conditions sufficient for the convergence of the empirical spectral distribution function of the Laplace matrix of a random graph to the free convolution of the standard normal distribution and the distribution of the semicircular law are formulated in terms of the probabilities of the presence of an edge and the variances of the corresponding weights. In terms of the Stieltjes transform, a characterization equation for the limiting distribution is obtained. The proof of the main result is based on the stability with respect to weak perturbations of the resulting characterization equation.
The recording of the report is available here.
November 16, 16:45 MSK
Andrei Zamyatin, Lomonosov Moscow State University
Regular dynamics of classical particles on a circle
The report is based on joint work with Professor V.A. Malyshev. A system of classical particles on a circle has been studied in the presence of a constant external force (the force depends only on time). Two types of dynamics have been investigated: dynamics with elastic collisions of particles and dynamics with a dissipative force acting on particles (in addition to an external force). For dynamics with elastic collisions, we have proved convergence (time tends to infinity) to a stationary flow, where particles move under the influence of deterministic or random forces. For dynamics with dissipation, we have given sufficient regularity conditions (conditions on the initial data and the parameters of the model) under which the particles do not collide with each other. Under the regularity conditions, we have proved the convergence (the number of particles tends to infinity) to a regular continuum system and we have given a rigorous derivation of the Euler equations for the limiting system. An explicit expression for the pressure have been also obtained.
The recording of the report is available here.
November 30, 16:45 MSK
Alexey Klimenko, Steklov Mathematical Institute of RAS, HSE
Convergence of spherical averages for actions of Fuchsian groups
Consider an action of a group G on a probability space by measure-preserving maps. Then for a function on this space we have an average over a finite subset M of the group, that is, a mean of all compositions of the function and the action of an element of M. If the group is endowed with a generating set O, then it is natural to consider spheres or balls in the group corresponding to the generating set O. The Cesaro convergence of the sequence of spherical averages has been obtained for a wide class of groups that are generated by a Markov coding (in particular, this includes all Gromov hyperbolic groups). However, the convergence of the spherical averages themselves was known for free groups only. We have shown that this convergence takes place for a wide class of Fuchsian groups by construction of new Markov codings satisfying some symmetry condition. The talk is based on the joint work with A. Bufetov and C.Series (arXiv:1805.11743).
The recording of the seminar is available here.
December 7, 16:45 MSK
Elena Zhizhina, ITTP RAS
Astral diffusion as the scaling limit of a symmetric random walk in a high contrast periodic environment.
We consider a symmetric random walk on the lattice in a high contrast periodic medium that can be interpreted as a discrete approximation of a diffusion with high-contrast periodic coefficients. From the existing homogenization results it is known that under diffusive scaling the limit behaviour of this random walk need not be Markovian, the effective evolution equation contains a term that represents the memory effect. We prove that the scaling limit of our random walk remains Markovian in some extended space. If in addition to the coordinate of the random walk we introduce an extra variable that characterizes the position of the random walk inside the period, then the limit dynamics of this two-component process, which we have called astral diffusion, will be Markovian.
References:
Piatnitski, A., Zhizhina, E., Scaling limit of symmetric random walk in high-contrast periodic environment, Journal of Statistical Physics, 169(3), 595-613 (2017)
A. Piatnitski, S. Pirogov, E. Zhizhina, Limit behaviour of diffusion in high-contrast media and related Markov semigroups, Applicable Analisys, 2019, Vol.98, No. 1-2, p. 217-231
The recording of the report is available here.
December 14, 16:45 MSK
Alexander Shen, LIRMM CRNS, Montpellier, France
Algorithmic information theory
Algorithmic information theory is somehow special: it is a mathematical theory (with its definitions, theorems and proofs) and at the same time it is motivated primarily by foundational questions about probability and statistics. I will try to discuss some of the following topics: Kolmogorov (algorithmic) complexity; randomness for finite and infinite objects; frequentism approach (Mises); martingale approach (Ville, Schnorr); game-theoretic approach (Shafer, Vovk) practical pseudorandom bit generators and testing; complexity theory and pseudorandomness laws of information theory are universal (Shannon, Kolmogorov, combinatorics); algorithmic statistics (Vitanyi, Vereshchagin).
The recording of the seminar is available here.
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