in Advanced Probability
"Random Walks and Approximation of Their Distributions"
April 02 - 07, 2017, Novosibirsk, Russian Federation
Novosibirsk Probability School, led by academician A.A. Borovkov, is well known in the scientific community both in Russia and abroad. Within the last three years, the Applied Probability Laboratory of NSU has successfully organised a number of research School in modern directions of probability theory, jointly with the Chair and Laboratory of Probability and Mathematical Statistics of NSU and Institute of Mathematics. In 2017, a number of similar Schools will be organised under the umbrella of the International Mathematical Center of NSU.
The second such School will take place 02-07 April. The School participants will have a chance to learn novel results obtained in the field of asymptotic analysis of distributions of functionals on random walks and also to understand the usage of such fundamental ideas of probability theory as the method of a single probability space and the method of replacing of probability measures.
The School will be opened on Sunday, April 2 at 14:30. The school's sessions will take place in room 4109, Novosibirsk State University.
The main purpose of the minicourse is to give an introduction to the filed of Markov chains with asymptotically zero drift. Such chains appear naturally in various stochastic models. Here we mention some of them: branching processes, risk processes with level-dependent premium rate, random polymers and excited random walks. The systematic study of such chains has been initiated by Lamperti in 1960's, but many exact asymptotic results have been obtained only very recently. In the first part of the minicourse we are going to discuss Lamperti's results on the transience, recurrence and positive recurrence of chains with asymptotically zero drift. Then we shall derive some limit theorems for the class of transient processes. In the second half we want to discuss the asymptotic properties of null recurrent and positive recurrent processes. Finally, we shall discuss some of the possible applications mentioned above.
The general goal of the minicourse is to introduce the students to some famous concepts and results related to the application of the method of a joint (single, common) probability space (or a coupling method). First, this is one of the most effective methods of probability theory, because it often gives simple solutions to very difficult problems (which sometimes are not solved by other methods). Secondly, it is simultaneously one of the most difficult (both for application and for understanding) methods of probability theory. In the first part of the course it is planned to disassemble in detail some simple examples of the application of this method, as well as to give with detailed explanations the famous fundamental theorems of Skorokhod, Strassen and Kantorovich-Rubinshtein. The second part of the course will be devoted to the history and state of affairs in a known problem related to the application of a number of modifications of this method for obtaining estimates of the rate of convergence in the Invariance Principle (Functional Central Limit Theorem). In particular, the method of Komlosh-Major- Tushnady and its generalizations will be discussed in detail.
|Novosibirsk State University|
|International Mathematical Center (IMC NSU)|