2022, Fall

Abstract: The report is devoted to continuous-time stochastic processes, which can be described in terms of reproduction, death and transport of particles. Such processes on multidimensional lattices are called branching random walks, and the points of the lattice at which the birth and death of particles can occur are called branching sources. Particular attention is paid to the analysis of the asymptotic behavior of particle numbers and their moments for symmetric branching random walks with finite branching sources and a finite or infinite number of initial particles under various assumptions about the variance of random walk jumps. The behavior of particle number moments is largely determined by the structure of the spectrum of an evolutionary operator of mean particle numbers and requires the use of the spectral theory of operators in Banach spaces to study of BRW models. The proof of some limit theorems on branching random walks with a finite number of sources and pseudo-sources, in which random walk symmetry breaking is allowed, is based on the verification of conditions guaranteeing the uniqueness of determining the limiting probability distribution of particle numbers by its moments. We also present results from the joint work of N.V. Smorodina and E.B. Yarovaya (Uspekhi Mat. Nauk, 2022), in which the proof of limit theorems is based on the approximation of the normalized number of particles at a lattice point by some non-negative martingale, which makes it possible to prove the convergence in the mean square of these quantities to the limit under fairly general assumptions on the characteristics of the process. 

The work is partly supported by the Russian Foundation for the Basic Research (RFBR), project No. 20-01-00487.

Video

Abstract: In the first talk we suggest introduction of relatively new random variable characteristic, Senatov moments, into the curriculum of probability theory courses. This suggestion is substantiated by tree viewpoints on the origin of Senatov's moments and their introduction will allow us to answer what can be considered as an analogue to Taylor series for density.

In the second talk we demonstrate practical application of Senatov moments.

Video

Abstract: Cumulative processes were introduced by Smith in 1955 and represent a rather wide class of random processes. This class includes, for example, compound renewal processes, the Cox processes, some types of L\'evy flights, integrals and partial sums of continuous- and discrete-time Markov chains, Markov modulated random walks.

Strong Gaussian approximation is a deep improvement of the classical invarian\-ce 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 (such as law of the iterated logarithm, arcsine law etc.), build consistent asymptotic variance estimates, analyze the limit behavior of various functionals in the approximated processes. The first such theorem was proved by Strassen in 1964. The Koml\'{o}s-Major-Tusn\'{a}dy  method providing the optimal approximation rates in the i.i.d. case attracted much attention and a lot of studies have been devoted to generalization of their bounds onto more general random systems.  In the talk we will present the  Komlos-Major-Tusnady  type theorems for cumulative processes and give applications to random sums with dependence on the summation index, integrated birth and death processes, non-Markov Levy flights, Markov modulated random walks.

Video

Abstract: Our object of study is the asymptotic growth of heaviest paths in a weighted (with signed weights) complete directed acyclic graph. Edge weights are i.i.d. random variables with common distribution F supported on [-\infty,1] with essential supremum equal to 1. Here weight -\infty means ``no edge''. The asymptotic growth rate is a constant that we denote by C(F). Even in the simplest case of two-point distribution F concentrated at 1 and at -\infty, corresponding to the longest path in the Barak-Erdos graph, there is no closed-form expression for this function, but good bounds do exist.

In this talk we present a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the weighted random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant C(F). Furthermore, we construct an effective perfect simulation algorithm for this functional .

The talk is based on a joint work with Takis Konstantopoulos, Bastien Mallein and Sanjay Ramassami.

Video

Abstract: Although Markov chains are widespread, and their stability and ergodicity conditions are well studied, the results on ergodicity for nonlinear Markov chains are quite limited. Such processes are nonlinear in the sense of the distribution law, i.e. the transition probabilities are dependent on both the current state and the current probability distributions of the process. New results on convergence estimates for nonlinear Markov chains will be discussed in the talk. Some results on limit theorems for nonlinear Markov chains will also be formulated, the law of large numbers and the central limit theorem.

Video, Slides

Abstract: Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called "Poisson Hypothesis". However, in most applications, this hypothesis is only conjectured. In this talk, we will present results on the validity of the Poisson Hypothesis for classes of processes in discrete and continuous time that include for example Galves-Löcherbach models from computational neuroscience.

Video, Slides