Concentration Phenomena In High Dimensions and Applications to Randomized Models

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This project addresses several challenging problems about high-dimensional probabilistic objects, that are related to the concentration phenomena. Part one is devoted to the study of probability distributions on linear spaces under certain convexity hypotheses of the Brunn-Minkowski kind. It focuses on general dimension free geometric and analytic properties of measures with heavy tails, expressed in terms of dilation, isoperimetric, and weighted Sobolev-type inequalities. The uniform distribution of mass in a convex body and more general log-concave distributions describe another important family in the hierarchy of convex or hyperbolic measures. As a closely related direction, the PI is also planning to consider some new aspects of the concentration phenomenon on product spaces of a large dimension. Part two deals with applications of different concentration phenomena to the randomized models, such as a partial summation of data, spectrum of stochastic matrices, etc., under minimal assumptions on the dependence of the observed random variables. Of a particular interest is an asymptotic behavior of typical distributions, resulting in a given randomized scheme. Part three is devoted to analysis of geometric characteristics of Markov kernels and associated random walks on graphs and other discrete structures. Discrete isoperimetric and modified forms of logarithmic Sobolev inequalities are planned to be considered in connection with the problem on the rates of the convergence of the Markov semi-groups. The study of the concentration phenomena is strongly dictated by various problems of Probability and Statistics on general global properties of stochastic processes. The Asymptotic Convex Geometry is another field, where a number of hard problems about high-dimensional convex bodies appeal to the concentration results and techniques. This study is also stimulated by problems in Combinatorics and Computer Science (such as computation of the volume) and in Mathematical Economics (optimization of the transport costs). The present proposal continues research in this direction and is aimed, in particular, to explore the role of the weak dependence in concentration phenomena, as well as its range of applicability in spaces of large dimensions.