UNITED SEMINAR OF THE DEPARTMENT OF PROBABILITY THEORY OF LOMONOSOV MOSCOW STATE UNIVERSITY
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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 spring 2023: professor Elena B. Yarovaya
Secretary of the seminar in spring 2023: Vladimir A. Kutsenko
To subscribe to the seminar newsletter or submit a talk, click "Subscribe or submit" at the top of the page.
We recommend you to study the network of seminars at this link, which were brought together by our colleagues from the Mathematical Center of the Southern Federal University.
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.
We recommend you to study the network of seminars at this link, which were brought together by our colleagues from the Mathematical Center of the Southern Federal University.
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.
Upcoming reports
Upcoming reports
Past reports
Past reports
October 4, 16:45 MSK
Albert Shiryaev, Moscow State University
On theories of decision-making under conditions of risk and uncertainty
The talk will describe paradoxes in game theory and an attempt to solve them using utility theory. The concepts of set capacity and Choquet integrals will be reviewed and their application to the solution of game paradoxes will be described. The introduced integrals will turn out to be a special case of nonlinear g-expectation, the meaning and properties of which will be discussed in the talk. In the final part the relations between Choquet integrals and backward stochastic differential equations for the model of incomplete financial markets will be considered.
October 11, 16:45 MSK
Vitalii Sobolev (Moscow Technical University of Communication and Informatics), Aleksandr Condratenko (Lomonosov Moscow State University)
Some aspects of studying queuing systems, finding convolutions of probability distributions in the central limit theorem and in other cases
This talk considers different types of asymptotic expansions in the central limit theorem (CLT), such as Edgeworth-Kramer and Gram-Charlier expansions, as well as those proposed by V.V. Senatov and V.N. Sobolev in the paper "On New Forms of Asymptotic Expansions in the Central Limit Theorem" (TVP, 2012). The latter allow us to obtain explicit estimates of the accuracy of the CLT-approximations any length. When expansions are constructed, there is a problem of increasing their accuracy. Here we will focus on the approaches proposed by X. Pravits (1991) and I.G. Shevtsova (2013) and and analyze them. Assuming the symmetry of the initial distribution, it is possible to obtain more accurate estimates presented by V.V. Senatov in 2016, and their generalizations proposed by V.N. Sobolev and A.E. Condratenko (TVP, 2023).
This is due to the fact that in these expansions, a part of their remainder is transferred to their main part, and this a priori guarantees better accuracy. The report will also show that it is possible to transform the expansions of I.G. Shevtsova, which allows us to construct asymptotic expansions in the same way in the general case. These CLT-expansions contain the last known moment of the original random variables in their main part. Applications of asymptotic methods in solving queuing tasks will be considered.
October 18, 16:45 MSK
Dmitrii Shabanov, MIPT, HSE
Probability thresholds for fractional colorings of random hypergraphs
The search for sharp probability thresholds for various properties is one of the central areas of study in the theory of random graphs and hypergraphs. A striking representative of the problems from this class is the famous RANDOM k-SAT problem on the feasibility of a random Boolean function. The symmetric variant of RANDOM k-SAT can be reduced to finding the probability threshold for the property of the existence of a proper coloring with two colors for a random k-uniform hypergraph. We will discuss a natural generalization of this problem concerning the so-called fractional colorings of hypergraphs. Using the second moment method and solving a number of extreme problems for doubly stochastic matrices, we were able to obtain very tight estimates of the probability threshold for the property of the existence of a fractional (4:2) coloring in the binomial model of a random hypergraph. Our results also show that this probability threshold is strictly greater than the probability threshold for the property of the proper 2-colorability. The talk is based on the joint work with P.A. Zakharov.
October 25, 16:45 MSK
Natalia Smorodina, Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
On a generalisation of the classical Ito formula
The talk is devoted to a generalisation of the classical Ito formula for a function from a Wiener process, in which the second derivative of the function is understood in terms of the differentiation of generalised functions. It is then shown how such a generalisation of the Ito formula is used for the investigation of integral functions from the Wiener process with a singular kernel.
November 1, 16:45 MSK
Andrei Lyulintsev, Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Markov branching random walks on Z+. Unlimited case
We consider a homogeneous continuous-time Markov process on the state space Z+, which we interpret as the movement of a particle. The particle can only transition to neighboring points in Z+, meaning that its coordinate changes by one unit with each change in position. The time spent by the particle at each point depends on its coordinate. The process is equipped with a branching mechanism. Branching sources can be located at each point in Z+, and we do not assume that the intensities are uniformly bounded. At the moment of branching, new particles appear at the branching point and continue to evolve independently of each other (and the other particles) according to the same laws as the initial particle. This branching Markov process corresponds to a Jacobian matrix. Formulas for the average number of particles in an arbitrary fixed point Z+ at time t>0 are obtained in terms of orthogonal polynomials corresponding to this matrix. The results are applied to some specific models, obtaining exact values for the average number of particles and finding their asymptotic behavior for large time.
November 8, 16:45 MSK
Grigorii Beliavsky, Natalia Danilova, Southern Federal University
Control in models with disorder. Binary interpretation
The report examines the control of the diffusion process with the parameters of drift and volatility changing at the moment of stopping. The report consists of four parts. The first part of the report contains a brief overview of the previous results and substantiates the actuality. In the second part, the problems of control processes with an “observable” disorder for symmetric and asymmetric criteria (mean-square and quantile hedging) are considered. The third part of the report discusses the “unobservable” disorder. In this case, a binary approximation of the basic process is considered, and the problem of estimating the decomposition on a binary sequence is solved. To reduce the amount of calculations, an algorithm for reducing the NP-complete problem to the P-complete problem is given. In the fourth part of the report, a binary discrete approximation of the Wiener process is considered, while the time interval is divided by passages of the absolute value of the Wiener process, which is more accurate than the standard approximation.
November 15, 16:45 MSK
Yury Khokhlov, Artem Volkov-Rarog, MSU
On a family of multidimensional distributions with heavy tails
The role of multidimensional normal distribution in many applied statistical studies is well known. This is due, in particular, to the fact that it has the following good properties. The density of this distribution is written out explicitly, which makes it possible to use the maximum likelihood method to estimate the parameters. Its characteristic function is also easily written out explicitly, which makes it easy to write out its numerical characteristics, for example, the average of each component and the covariance matrix. Any linear combination of coordinates will be a distribution of the same type. This makes it easy to find the distribution of a single coordinate, the distribution of the sum of coordinates, the conditional distribution of one coordinate provided for the sum of all coordinates. But the tails of such a distribution decrease very quickly, which is often not done in many specific tasks. In our report, we consider a certain family of multidimensional distributions that has all the properties listed above, but, unlike a multidimensional normal distribution, the tails of such distributions decrease in a power-law manner, i.e. they are heavy. Some variant of such family has been considered earlier. But our definition is more general and we investigate it using a new method when these distributions are considered both as densities and as characteristic functions at the same time. In addition, their close connection with limit distributions for multidimensional geometric random sums is shown.
November 22, 16:45 MSK
Artem Kovalevskii, S. L. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences. S. L. Sobolev Institute of Mathematics SB RAS; Novosibirsk State University; Novosibirsk State Technical University
Functional central limit theorem for processes of numbers of different words in forward and backward directions in an elementary probabilistic model of text
An elementary probabilistic model of a text assumes that words of the text take values in an infinite dictionary independently of each other in accordance with some discrete probability distribution. According to the generalized Zipf law, this distribution is regularly varying. Under these assumptions, we prove a two-dimensional Functional Central Limit Theorem for processes that count the number of different words along a text in the forward and backward directions. As a consequence we get Functional Limit Theorem for the difference between these processes and calculate the limit distribution of the normalized integral of its square. Based on this research, we construct a statistical test for text homogeneity and demonstrate its application on real texts.
November 29, 16:45 MSK
Ekaterina Vl. Bulinskaya, MSU
Probabilistic-geometric properties of a spatial branching random walk
In the talk, we consider our main results published in a series of 15 papers (without co-authors) over the past ten years after defending a PhD thesis. The conducted research belongs to the intensively developing held of interrelated problems for Markov chains, branching processes and branching random walks. The list of cited literature contains more than 120 items. We only mention the works by S.Albeverio, L.V.Bogachev, E.B.Yarovaya (1998) and V.A.Vatutin, V.A.Topchij, E.B.Yarovaya (2003) that are at the origin of many stochastic models taking into account the mechanisms of random movement of particles and their branching. The main feature of the cycle is the construction of a comprehensive picture of the propagation, with probability one (in contrast to the works by S.A.Molchanov and E.B.Yarovaya), of the population front for catalytic branching random walk in a space of any finite dimension where a finite set of catalysts has an arbitrary configuration. To solve these tasks, a combination of probabilistic techniques and analytical apparatus was required. The proofs of the main results of the considered series of papers use auxiliary multi-type Bellman-Harris processes, the hitting times under taboo (that arose in the works by K-L.Chung and A.M.Zubkov) for Markov chains, multidimensional renewal theorems and spinal technique (many-to-one lemma), which is a modern tool in the theory of branching processes. Moreover, Tauberian theorems, the theory of determinants, the Laplace transform, as well as convex analysis, martingale change of measure, the theory of large deviations and the coupling method are used as well.
December 06, 16:45 MSK
Vasilii Kolokoltsov, MSU, HSE
Domains of quasi-attraction of stable laws
In the talk I like to advertise three groups of new results in the three related new directions of research: 1) Rates of convergence in the functional CLT with stable limits; 2) Rates of convergence of CTRWs (continuous time random walks) to fractional evolutions; 3) A solution to the long-standing question on using stable laws in reality: why stable laws can be met in practice, while everything around us is finite (and thus has finite moments)? We introduce domains of quasi-attraction as distributions, whose normalised sums of n i.i.d terms approach stable laws for large, but not too large n. We supply full quantitative and qualitative description of this effect, which is already well in use by physicists. The ideas of the talk are taken from the recent author's papers (1) The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions. Fractal Fract. (2023), 7, 335. (https://doi.org/10.3390/fractalfract7040335); (2) Domains of Quasi Attraction: Why Stable Processes Are Observed in Reality? Fractal Fract. (2023), 7, 752. (https://doi.org/10.3390/fractalfract7100752)
December 13, 16:45 MSK
Yuliana Linke, MSU, Sobolev Institute of Mathematics
Universal kernel type estimates in nonparametric regression with applications to nonlinear regression models
The report is devoted to estimation methodology in both nonparametric and nonlinear regression problems in the case of so-called dense data. For wide classes of regression models, new universal uniformly consistent kernel-type estimators are proposed in the following classical problems of nonparametric regression: estimating a regression function from observations of its noisy values in a certain known set of points from its domain of definition, called regressors, as well as estimating the mean and covariance functions of a random process in a scheme where each of the independent copies of the process is observed in a noisy version in one or another set of regressors. In numerous papers by predecessors, the regressors are assumed to be either fixed and regularly filled the domain of definition of the function, or random and consisted of independent or weakly dependent random variables. The fundamental novelty of the presented results lies in the ability to construct uniformly consistent estimators in the absence of any information about the nature of the dependence of the regressors. For example, in the case of estimating a regression function, only the condition is assumed that the regressors densely fill the domain of definition of the regression function. This condition is insensitive to the correlation of regressors, essentially is necessary to restore the function with some accuracy and includes both the situation of fixed regressors without the additional requirement of regularity, and random regressors not necessarily satisfying the weak dependence conditions. The concept of dense data is also implemented in various formulations of the problem of estimating both the mean and covariance functions of a random process. Аs an application, we consider two approaches to solving the problem of constructing preliminary estimators of finite-dimensional parameters in nonlinear regression models in the case of dense filling a certain area with regressors and without the requirement of complete control over them. Preliminary estimators play an important role in optimal parameter estimation in nonlinear regression problems. Previously, preliminary estimators were found only for a small number of nonlinear regression models, and the problem of constructing them for fairly wide classes of nonlinear regression models remained open.
December 20, 16:45 MSK
Vladimir Kutsenko, MSU
Effects of random environment in processes with generation and transport of particles on lattices
The talk presents the author's main results from five papers on branching random walks in random environment. Apparently a perturbation of the difference Laplacian by a random potential was firstly considered in a series of papers by Ya. Zeldovich and co-authors in the 80s of the last century. Such models were further developed and formalized in the works of J. Gertner and S. Molchanov in the 1990s. S. Albeverio et al. (2000) and E. Yarovaya (2012) have already considered branching random walk on a multidimensional lattice with continuous time in a random environment. We continue the study of such branching random walk. In the first part of the talk, the random environment at each point of the lattice is defined by non-negative, independent and identically distributed random intensities of particle multiplication and death. In this sense, we can speak of the homogeneity of the random environment. We present a series of results completing the study of local particle numbers averaged over the environment for the case of an asymptotically Gumbel potential. In the second part of the paper, in contrast to previous studies, an non-homogeneous branching medium is considered. At one point of the lattice, the multiplication of particles with constant intensity is allowed, while at the other points only the death of particles with random intensity is possible. A branching random walk on a one-dimensional lattice and the case of a bounded potential are considered. For such a model, non-averaged over the environment average particle numbers are investigated. We applied the Laplace method for integrals, Feynman-Katz type representations, resolvent path expansion, and methods for the study of the random spectrum of the evolution operators of the mean particle number. The last part of the paper presents the results of numerical simulations, which most likely have not been performed before for branching random walks in random environment.
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