2021, Fall

Abstract: We consider a non-homogeneous compound renewal process. We suppose that the  values ​​of  its jumps are independent  and satisfy only the classical Lindeberg condition, but the magnitude of each such jump may depend on the random distance to the previous jump. In physical literature such processes are often called random walks with continuous time. We are interested in the exact asymptotics for the probability that on an increasing time interval the process under study will stay above a certain non-constant boundary. For walks with nonrandom distances between jumps, this problem was solved earlier by the author together with V. Wachtel and D. Denisov. The present research was carried out jointly with A. Shelepova.

Video, Slides

Abstract: joint work with S.N. Evans (Berkeley).

The family of metric measure spaces can be endowed with the semigroup operation being the Cartesian product. The aim of this talk is to arrive at the generalisation of the fundamental theorem of arithmetics for metric measure spaces that provides a unique decomposition of a general space into prime factors. These results are complementary to several partial results available for metric spaces (like de Rham's theorem on decomposition of manifolds). Finally, the infinitely divisible and stable laws on the semigroup of metric measure spaces are characterised.

Video, Slides

Abstract: The talk is devoted to a distribution of a maximal value in a finite sample from a GEM distribution. The GEM distribution is a random partition of the unit interval; its distribution is parametrised by two parameters \alpha and \theta. This random partition and its properties will be described in the talk. One can also interpret it as a random discrete distribution on natural numbers. A sample from it is exchangeable. In the simpler case \alpha=0 we are able to describe a distribution of a maximum of n such random variables as a sum of n independent geometric random variables. In the more tricky case \alpha>0 such representation, perhaps, does not exist. Yet we can show that the maximum of n samples behaves asymptotically as n^{\alpha/(1-\alpha)} up to a random factor whose distribution is explicitly described.

The talk is based on the joint works of the author with Jim Pitman.

Video, Slides

Abstract: In the talk we consider one-dimensional stochastic flows which consist of the ordered Brownian motions starting from the points of the real line. Such flows are uniquely defined by its mutual covariance function. In case when this function is smooth the flow can be constructed as the flow of solutions to Cauchy problem for stochastic differential equation. For non-smooth or discontinuous covariance function the additional construction is necessary and coalescence of the particles can occur. Convergence of covariance functions leeds to the convergence of n-point motions of the flows. This causes the convergence of the images of Lebesgue measure under the flows in the corresponding distances. We discuss the speed of such convergence and the behaviour of the total time of free motion for the particles in the flow and the quadratic entropy for the sets of values of the flow.

Video

Abstract: Limit theorems for superposition of independent point processes (PPs) must involve an operation that makes them “thinner” so that a limit of superposition of their growing number exist. It is analogous to scaling for random variables, but to preserve a PP framework, this “scaling” of a PP must be stochastic acting independently on the PP’ points. We show that the most general such operation on a PP is independent subcritical branching of its points. The easiest example is given by pure-death process that is equivalent to independent thinning of points. Given such an operation, one can formulate limit theorems for superposition of independent PPs aiming to characterize all possible limits. The processes which may arise as a limit are selfdecomposable (SD) PPs which are a strict subclass of infinitely divisible (ID) PPs. At the same time, it is strictly larger than the class of Strictly stable PPs which arise as a limit of scaled superposition of i.i.d. PPs.  Since SD PPs are also ID, their distribution is characterized by Levy measure (also known as KLM measure in PP context) and it has a special integral representation from potential theory and the theory of general Markov processes.

Video, Slides

Abstract: A random graph in the binomial model G(n,p) (a random graph in the Erdős-Renyi model) has been one of the main objects of study in probabilistic combinatorics since the end of the 50-s of the past century. One of first questions posed by P. Erdős was a question concerning the asymptotic behavior of the chromatic number of the random graph G(n,p). In 1991 T. Luczak proved that for not fast growing product np, the chromatic number of the random graph is concentrated in two consecutive numbers, which however were unknown. We will present our recent results where these values have been found for almost all functions p=p(n) up to o(n^{-3/4}).

Video, Slides

Abstract: Complex-valued stable random variables with a complex stability index are constructed. It will be shown that the obtained stable values satisfy the usual stability condition, but for a complex parameter. At the same time, it will be proved that no other distributions can satisfy such a stability condition. The characteristic function of the obtained random variables is found and it is shown that the distribution is infinitely divisible. Limit theorems for sums of complex-valued iid random variables are shown and Levy processes are constructed. Using the obtained Levy processes, a semigroup of operators is constructed and its generator is found.

Video, Slides

Abstract: We consider a multidimensional random walk in a cone. In the finite-variance case under some conditions we will discuss construction of  harmonic  functions for random walks killed on exit from the cone and corresponding asymptotics for exit times. Some example will illustrate that the conditions are close to minimal.  This is a joint work with V. Wachtel which continues  arXiv: 1805.01437<https://arxiv.org/abs/1805.01437> and arXiv: 1110.1254<http://arxiv.org/abs/1110.1254>.