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

May 14, 16:45 msk

Elena Lenena, Lomonosov MSU, Russia

Functional Limit Theorems for Queueing Networks in a Random Environment

The dissertation investigates generalized Jackson networks with regenerative input flows in a random environment. The main focus is on proving functional limit theorems: it establishes a strong approximation of the queue length vector by reflected Brownian motion in the positive orthant and derives explicit formulas for key parameters such as drift, reflection matrix, and covariance. The work provides estimates for deviation probabilities and the Wasserstein distance between the distributions of the processes. A method for estimating the covariance matrix in overloaded systems is developed. The dissertation addresses both theoretical aspects and practical examples, including transportation networks with unreliable components. The results are applicable to the analysis of stability, optimization, and diagnostics of complex service systems.

The recording: YouTube

April 16, 16:45 msk

Tatyana Turova, Lund University, Sweden

Antiferromagnetic Covariance Structure of Coulomb Chain

We consider a system of particles lined up on a finite interval with 3-dimensional Coulomb interactions. Typically, models with interactions between all pairs of particles are studied. The main question of interest is the distribution of spacings between the consecutive particles, and the scaling of the moments with respect to the number of particles when the latter goes to infinity. Malyshev (2015) suggested to include in the model only the interactions between the closest neighbours to study the flow of charged particles. Notably, even the nearest-neighbours interactions case is proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Assuming zero external force, we show that interactions beyond the nearest ones lead to qualitatively new features of the system. In particular, we discover that the covariances between spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of the number of spacings between them, while their amplitude decays. In the course of the proof of these results, a conditional Central Limit Theorem for dependent random variables is established which might have interest on its own (particularly due to negatively correlated variables). As a corollary conditional Central Limit Theorem is derived which confirms that the fluctuations in the considered ensemble are Gaussian.

The recording: YouTube

April 2, 16:45 msk

Lomonosov readings

Elena Filichkina, Lomonosov MSU, Russia

On non-Markov processes satisfying the Kolmogorov-Chapman equations

The report considers examples of processes that satisfy the Kolmogorov-Chapman equations, but are not Markovian. In particular, the example proposed by Feller of a process with three or more states is analyzed in detail, and examples of processes with two states are considered, based on Bernstein's example of pairwise independent variables which are not mutually independent. It is shown that the Kolmogorov-Chapman equations do not uniquely determine a non-Markov process. And it is established that for non-degenerate Gaussian processes with a continuous covariance function, the satisfying of the Kolmogorov-Chapman equations is equivalent to the Markov property of the process.

Alexander Veretennikov, Lomonosov MSU, Russia

On new limit theorems and applications of limit theorems

A review of fresh results on LLN, CLT and some applications will be offered. The following results will be presented: theorems of strong, weak LLN, and LLN on a sublinear probability space; Dobrushin's type CLTs for non-homogeneous Markov chains; application of Bernoulli LLN to convergence of mixed derivatives of Bernstein type polynomials for functions of two variables, which is an analogue and expansion of a Kantorovich result for one-dimensional case.

The recording: YouTube

March 19, 16:45 msk

Andrey Piatnitski, MIPT, Russia; UiT, Norway

On spectral properties of convolution operators with potential

The talk will focus on the spectral properties of the operator being the sum of a convolution operator and multiplication by a potential. Under the assumption that the potential is a real bounded function that tends to zero at infinity, and the convolution kernel is an even integrable function this operator is bounded and symmetric in L^2. We will describe both the essential and the discrete spectrum of this operator.

The recording: YouTube

March 5, 16:45 msk

Yuri Yakubovich, St. Petersburg University, Russia

Different cluster orderings in Gibbs random samples

We consider samples from random discrete distributions and the corresponding exchangeable random partitions of countable sets. After recalling a general theory of exchangeable random partitions due to Kingman, we introduce a two-parametric Ewens family of exchangeable random partitions and its generalization, the so-called Gibbs model of random partitions. We are interested in various orders of clusters of exchangeable random partition of a countable set restricted to its finite subset. There are at least three natural orderings of blocks of random partitions: decreasing cluster sizes/frequencies, appearance order and value order, to be explained in detail during my talk. For the most studied one-parametric Ewens family it is known from works of Donnelly and co-authors that the appearance order and value order have the same distribution. I will explain that for a two-parametric generalization this is no longer the case, and describe an explicit procedure for passing from one order to another. This procedure works as well for the Gibbs frequencies in size-biased random order. The talk is based on joint work with Jim Pitman.

The recording: YouTube

February 19, 16:45 msk

Andrei Lyulintsev, St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia

Jacobi branching random walks corresponding to orthogonal polynomials of discrete variable

A branching random walk on Z_{+} is considered, which corresponds to a Jacobi matrix. Previously, formulas for the average number of particles at an arbitrary fixed point in Z_{+} at time t > 0 were obtained in terms of the orthogonal polynomials associated with this matrix. In the present work, the application of the obtained results to certain models involving orthogonal polynomials of a discrete variable (Krawtchouk, Meixner and Poisson-Charlier polynomials) will be discussed.

The recording: YouTube