UNITED SEMINAR OF THE DEPARTMENT OF PROBABILITY THEORY OF LOMONOSOV MOSCOW STATE UNIVERSITY
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This is the page of the spring semester of 2023 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.
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.
Upcoming reports
Upcoming reports
Past reports
Past reports
February 8, 16:45 MSK
Egor Illarionov, Dmitrii Sokoloff, Moscow State University, Moscow Center of Fundamental and Applied Mathematic
Relative efficiency of the three mechanisms of vector fields growth in a random media
We consider a model of a random media with fixed and finite memory time (short-correlated model). Inside the intervals of the constant parameters of the media, we can observe either amplification or oscillation of a vector field in a given particle. Cumulative effect of amplifications in many subsequent intervals clearly leads to amplification of the mean field and mean energy. Similarly, the cumulative effect of intermittent amplifications or oscillations also leads to amplification of the mean field and mean energy, however, at a lower rate. Finally, it is also possible that the random oscillations alone will resonate and yield the growth of the mean field and energy. These are the three mechanisms that we investigate and compute analytically and numerically the growth rates on a basis of the Jacobi equation with the random curvature parameter
February 15, 16:45 MSK
Pavel Tesemnikov, Sobolev Institute of Mathematics of SB RAS, Novosibirsk State University
Asymptotic analysis of heavy-tailed random walks on stochastic ordered graphs and related problems
We study asymptotic properties of two types of random walks on stochastic ordered graphs. In the first part of the talk we show that the so-called «principle of a single big jump» (PSBJ) holds for a random walk on a genealogical tree of a branching process in varying environment, where the increments of the random walk follow a heavy-tailed distribution with infinite exponential moments. We obtain the exact tail asymptotics for the maximum of the random walk over a random (possibly infinite) number of generations. In the second part of the talk we consider a random walk on a generalised Barak -- Erdos graph (a directed version of the Erdos -- Renyi random graph) where an edge is present with probability that depends both on the vertices incident to the edge and on the total number of the vertices in the graph. Assuming again the heavy-tailedness of the increments of the random walk, we obtain the exact tail asymptotics for the maximum of the random walk along a randomly chosen path of minimal length that connects the extremal vertices. We show that this asymptotics agrees with the PSBJ again. Further, we study the limiting distribution of the minimal path length between extremal vertices as the total number of vertices in the graphs growths. The talk is based on joint papers with Sergey Foss and Bastien Mallein.
February 22, 16:45 MSK
Aleksei Lebedev, MSU
Anastasiia Gorbunova, Institute of Control Sciences
Nontransitive systems of continuous random variables and their applications
The problem of nontransitivity of the stochastic precedence relation for sets of random variables with distributions from certain classes is studied. Initially, this question was posed in connection with a problem from the theory of strength (Trybula, 1961). In the future, the topic of nontransitivity became popular on the example of the so-called nontransitive dice. The paper presents criteria by which it is proved that nontransitivity cannot exist for many classical continuous distributions (uniform, exponential, normal, logistic, Laplace, Cauchy, Simpson, one-parameter Weibull, etc.). Next, we consider the case of distributions with polynomial density on the unit interval, where nontransitivity can occur for a sufficiently large degree. A method for constructing nontransitive triplets of mixtures of distributions is presented, which works for mixtures of normal and exponential distributions. The theme of the influence of nontransitivity on the behavior of stochastic systems is touched upon, i.e. how different relationships between random variables at the input of stochastic systems affect the output. The effects of nontransitivity in queuing systems are studied in two models: when the service times in different systems form nontransitive sets and when the parameters of the queuing systems are random.
March 1, 16:45 MSK
Ivan Alexeev, MSU
Discrete quasi-infinitely divisible random vectors
Multivariate discrete probability laws are considered. In this talk it will be shown that such laws are quasi-infinitely divisible if and only if their characteristic functions are separated from zero. We generalizes the existing results for the univariate discrete laws and for the multivariate laws on integer lattice. Similarly to vectors on an integer lattice, the Cramer-Wold device for quasi-infinite divisibility and infinite divisibility for discrete random vectors will be presented.
March 15, 16:45 MSK
Stanislav Molchanov, UNC Charlotte, HSE
Discrete dynamo
One of many applications of the famous Kolmogorov’ theory of the isotropic turbulence is related to the problem of the kinematic dynamo in magnetohydrodynamics. This problem was popular in the physics literature in the years 1960th - 1990th. Many results in this area are going to Ya. Zeldovich and his group. Mathematically, it is a question about the evolution of the magnetic field of the stars, i.e., about the qualitative behavior of the solutions of the Maxwell equation in the turbulent flow of the conducting fluid. The well-known review by Ya. Zeldovich, S. Molchanov, A. Ruzmaikin and D. Sokolov contains a description of the important phase transition. The second moment of the magnetic field in dynamo model is exponentially increasing for the hot stars and decreasing for the low-temperature stars. This result cannot guarantee a.e. increasing of the field (due to the intermittency effects). The talk will present a simplified discrete model which demonstrates a.e. phase transition.
March 22, 16:45 MSK
Andrei Piatnitskii, ITTP RAS
Large deviations for Markov jump processes in locally periodic environments
The talk will focus on the large deviation principle for a family of jump Markov processes defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems. It is shown that the corresponding Lagrangian is a convex function of velocity that has a superlinear growth at infinity. However, neither the Lagrangian nor the corresponding Hamiltonian need not be strictly convex, we only claim their strict convexity in some neighbourhood of infinity.
March 29, 16:45 MSK
Elena Bashtova, MSU
On almost sure approximation for some types of multidimensional random walks
Some studies of particles (bacteria, leukocytes, etc.) movement show that it can be described by random piecewise-linear functions. A particle moves along a straight segment at a constant rate, and then selects the turn direction randomly. K. Pearson was the first to pose the question about the properties of such a process, called later a random flight. Formalization of these and other observed phenomena leads to different types of random walk such as Levy flights, Markov-modulated random walks etc. It is worth noting that many natural modifications of a random walk turn out to be inhomogeneous or non-Markovian. Strong Gaussian approximation is a deep improvement of the classical invariance principle, as it provides a way to construct a Wiener process with trajectories almost surely close to the ones of the given random process. Such approximation allows to obtain other limit theorems, build consistent estimates of the asymptotic variance, analyze the limit of various functionals depending on the approximated processes. The talk will present strong Gaussian approximation theorems for Markov-modulated random walks and some non-Markovian random walks with nonstationary increments.
April 5, 16:45 MSK
Jordan M. Stoyanov, Bulgarian Academy of Sciences and Shandong University, China
New characterizations of the normal and gamma distributions by using independence of two statistics and Anosov’s theorem
Among the distributions on the whole real line and on the positive half line, the normal distribution (N) and gamma distribution (Г) play a significant role. They are well studied which is good for the theory and the applications. Available in the literature are diverse characterizations of both, including via independence of two appropriately chosen statistics, say A and B. Well–known are the classical results: for N, when A is the sample mean and B is the sample variance; for Г, A is the sample mean, B is the sample coefficient of variation. Many years ago, D.V. Anosov, with a reference to Yu.V. Linnik, suggested to use (nontrivial) integro–functional equation in order to prove a new characterization of N. Several years later, Anosov’s method was essentially extended, so new characterizations were obtained for N. Moreover, Anosov’s theorem and his integro–functional equation for N were further studied and an analogue found as a suitable tool to study Г. This is how we have today a reasonable number of new characterizations of both N and Г. Some general results will be reported, they are based on independence of the statistic A, the sample mean, while the second statistic, B, is now chosen from a big class consisting of admissible homogeneous statistics. This class is defined in terms of the sample order statistics. There is an interesting parallel between the new results in the two cases, N and Г. Of a special interest is a series of corollaries which are not only new, but they are given in common terms, e.g., as the sample range, or Gini’s coefficient. A few challenging open questions will be outlined. The results to be reported come out after years of joint work of the speaker with G.D. Lin and C.–Y. Hu (Taipei). There are two papers, one on N, already published in AISM, Tokyo (Springer 2022), the second one, on Г, accepted for publication (2023).
April 12, 16:45 MSK
Lomonosov readings
Ekaterina Bulinskaya, MSU
Discrete-time insurance models
Input-output mathematical models are appropriate for investigation of real processes arising in such applications as insurance, finances, inventory, queueing, reliability, population dynamics and many others. We formulate the results in terms of insurance being the oldest application field of probability theory. Our main goal is to establish an optimal control providing the extremum of a chosen risk measure (objective function). The class of admissible controls includes the company initial capital, tariffication, coinsurance and reinsurance, bank loans and investment. One has to be sure that the model under consideration is stable to use the obtained results. We also consider the limit behavior of the company capital as the planning horizon tends to infinity and prove strong law of large numbers and central limit theorem. Several discrete-time models are studied since in some cases the describe more precisely the real situation, they are convenient for numerical calculations, and can also be used to approximate the corresponding continuous-time models.
Aleksandr Veretennikov, IITP RAS, MSU
On strong law of large numbers
A new remark on a version of a strong law of large numbers for a sequence of pairwise independent random variables is proposed. The main goal is to relax the assumption on the existence of a finite expectation for each of the summands. Some historically important, although less known papers on the topic will also be mentioned. The talk is based on the joint preprint with Alina Akhmiarova.
Vladimir Kutsenko, Dmitry Sokoloff, Elena Yarovaya, MSU
Peaking regimes in branching random walks with random particle generation intensities
We consider the evolution of a particle system on a multidimensional lattice with continuous time. At the initial moment of time, there is one particle in the system that can split into two, die or move to a neighboring point of the lattice. The evolution of particles takes place in a random medium, i.e. the intensities of reproduction and death of particles are set by stationary random variables. New particles evolve independently of each other and the entire prehistory. In this system, the growth of the average number of particles is studied, which depends on the difference between the intensity of splitting and the intensity of death, called the random potential. The fundamental foundations for the study of such models are laid in the works of Ya. Zeldovich, S. Molchanov, J. Gärtner, W. König and co-authors. Such models are mainly used in statistical physics and in studying the dynamics of various populations. We have shown that if the potential decreases slowly enough at infinity, then there is an explosive increase in the number of particles and their average number can go to infinity immediately after the beginning of the evolution of the system. In addition, the following result is proved: if the average number of particles for each specific implementation of the medium is finite, then this condition does not guarantee that the average number of particles will remain finite when averaged over all implementations of the medium. Finally, the behavior of medium-averaged moments of particle numbers for asymptotically Gumbelian potentials at long times is described.
April 19, 16:45 MSK
Aleksey Polunchenko, Binghamton University, USA
New bounds for the quasi-stationary distribution of the Shiryaev martingale with an application to quickest change-point detection
We consider the quickest change-point detection problem for Brownian motion under Pollak's (1985) minimax setting. It was recently shown by the author (TPA 2017) that the randomized Shiryaev-Roberts-Pollak procedure, whose decision statistic is based on the Shiryaev martingale, is nearly Pollak-minimax in the limit, as the risk of a false alarm goes to zero. The procedure is 'randomized' because the initial value of the decision statistic is random, and is sampled for the quasi-stationary distribution of the Shiryaev process. The near minimaxity of the procedure was shown using the trivial upper bound on the cdf of the quasi-stationary distribution. In this work we obtain new, tight lower and upper bounds for the quasi-stationary distribution; the bounds are obtained using the latest (2022) results from the theory of Bessel functions. The new bounds for the quasi-stationary distribution allow us to accurately estimate the rate of convergence of the detection delay delivered by the Shiryaev-Roberts-Pollak procedure to the (unknown) optimum.
May 03, 16:45 MSK
Maksim Savelov, MSU
The limit joint distributions of statistics used to test the quality of random binary sequences generators
The NIST Statistical Test Suite developed by the National Institute of Standards and Technology (USA), is one of the most popular tools designed to solve the important practical problem of testing the quality of random binary sequence generators. The report contains results on the limiting joint distributions of various sets of NIST test statistics under the hypothesis that the tested sequence is a Bernoulli one. Necessary and sufficient conditions for asymptotic uncorrelatedness and necessary and sufficient conditions for asymptotic independence of considered statistics will be discussed. Our results may be used to choose the parameters of the NIST Statistical Test Suite.
May 10, 16:45 MSK
Aleksandr Veretennikov, IITP RAS, MSU
Ergodic properties and ergodic coefficients for Markov chains (based on joint works with Oleg Butkovsky, Maria Veretennikova, and Alexander Shchegolev)
A.A. Markov himself introduced a coefficient for studying convergence of Markov chains (MC) with finite state spaces towards the stationary regime; much later this characteristics was called Dobrushin's ergodic coefficient. This is why the author prefers to call it Markov– Dobrushin's (MD in what follows), although, even more correct name might be Markov–Kolmogorov–Dobrushin's. The reasons for all of this will be givenin the talk. In attempts to find a better effective bound for the rate of convergence, the author of this talk introduced a new characteristic in a series of recent papers: the spectral radius of a certain sub-Markov operator related to the so called markovian coupling. Examples show that the new bound is no worse than that guaranteed by the MD coefficient, and in most cases it is better. The new approach also works for some nonhomogeneous MC (as well as the analogue of the MD coefficient). It is yet known the result by Gantmacher that for homogeneous MC the best unimprovable bound is provided by a certain exponentially decreasing function with the exponential index coinciding with the log of the modulus of the second eigenvalue of the transition matrix or operator. In the talk the links between the new coefficient and this unimprovable bound will be shown: namely, it turns out that taking the analogue of the sub-Markov operator mentioned earlier for several steps, it is possible to approach the unimprovable estimate arbitrarily close. Notice that for non–homogeneous MC this second eigenvalue approach does not work at all. Finally, it will be shown how the new coefficient may be applied to estimate the rate of convergence for certain classes of nonlinear MC chains by the method of small perturbations.
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