2022, Spring

Abstract: We study the joint asymptotics of  forward and backward processes of  numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. The corollary of the main theorem asserts the weak convergence of the integral of the difference of forward and backward  processes to the normal distribution. We obtain parameter estimates that have a joint normal distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn scheme on the number of thrown balls. We analyse the statistical tests by simulation and apply them to the analysis of the homogeneity of texts in natural language.

Video

Abstract: Back in the 30s of the last century, P. L\'evy proved that $\alpha$ stable random variables and only they are limits for the sums of i.i.d. random variables with positive normalization and some centering. Later, Feldheim generalized this result to the case of random vectors. Namely, he proved that $\alpha$ stable random vectors and only they are limits for sum i.i.d. random vectors with positive normalization and some vector centering. This report will answer the question about the limit laws for the scheme of summation of i.i.d. complex-valued random vectors with complex normalization and some vector centering.

Video

Abstract: We consider a general allocation scheme of n particles by N cells, defined by independent random variables, which have power series distribution with the parameter β. Let D[0, 1] be a Skhorohod space. Suppose the random processes X_{n,N}(t), 0 <= t <= 1, is a number from particles in the first [tN] cells.

We prove that under some conditions if n is a fixed number and N → ∞ then X_{n,N} converge to n F_n in D[0, 1], where F_n is an empirical process. Also we prove that under other conditions centered and normed random processes X_{n,N}(t), as n, N → ∞, converge to a Brownian bridge in D[0, 1].

Video

Abstract: A review of the results on the stability of the transition densities of some classes of Markov chains and diffusions with respect to perturbations of the coefficients will be made. An estimate of the error in various metrics will be given and possible applications of such results will be indicated. We will also discuss proofs, the main tool of which is the parametrix method.

Video

Abstract: We will discuss how to predict the future, to write several books, to become rich and even be invited to the Oprah Winfrey Show.

Video, Slides

Abstract:  This talk will present an approach to asymptotic analysis, with respect to observation time horizon and noise intensity, of solutions to linear filtering and parameter estimation problems for processes such as fractional Brownian motion. A similar technique is applicable to the problem of finding exact asymptotic approximations of the eigenvalues and eigenfunctions of the corresponding covariance operators.

Joint work with Pavel Chigansky (University of Jerusalem) and Dmytro Marushkevych (University of Copenhagen).

Video, Slides

Abstract: We obtain the exact tail asymptotics for the rightmost point of the branching random walk with fading whose increments have a heavy-tailed distribution. In particular, we show that the result obtained agrees with so-called «principle of a single big jump», which is typical for the model with heavy-tailed characteristics. The talk is based on a joint work with Sergey Foss (to appear in Proceedings of the Steklov Mathematical Institute, Vol. 316).

Video, Slides

Abstract: We discuss some problems associated with  probabilistic representation of the resolvent operator.

Video