This proposal focuses on the study of geometric, analytic and information-theoretic aspects of high dimensional phenomena on the border of probability, convex geometry and analysis. One part of the project concerns the problem of rates of convergence in the entropic central limit theorem, and is devoted to obtaining new asymptotic expansions for the relative entropy with respect to the growing dimension. In other part, it is proposed to perform a systematic study of the dimensional behavior of the entropy and information for different classes of probability distributions, satisfying convexity conditions. In particular, new concentration properties of the information content will be considered for dependent high-dimensional data. It is planned to introduce and explore special positions of probability measures, responsible for correct behaviour of sums of independent summands (when the entropy power inequality can be reversed). Another part addresses the stability problem, raised by Kac and McKean, in the entropic variant of Cramer's characterization of the normal law.
The main theme of the proposal is the development of the information-theoretic approach to high dimensional phenomena, with focus on obtaining new asymptotic bounds on the entropy and information. The study of entropy is dictated by various applications within and beyond pure mathematics. Entropy plays a key role in statistical physics (in order to capture the amount of disorder in a system), in statistics (to measure the performance of statistical estimators), in engineering and mathematical theory of communication. The proposed research also aims to provide new connections between probability, geometric functional analysis and information theory, and to demonstrate an increasing role of entropy bounds in purely mathematical fields.
An integral component of the project is the involvement and training of the graduate and undergraduate students.