Quenched Tails and Almost Sure Limit Laws

Stanford University, Stanford CA

Investigators

Abstract

0072331 Dembo This research is in the area of large deviations and its applications. In particular, the PI will study situations where the rare behavior conditioned on a fixed (= quenched) realization of one element of randomness is the key to determine and understand the typical phenomena. One unifying theme is that the evaluation of probabilities of rare events leads to an understanding of the mechanisms by which they can occur, and when many different rare events are possible, it leads to the identification of the mechanism causing one of them to actually occur. Two specific lines of research are planned: (i) Mathematical study and understanding of "aging" phenomenon in the large time behavior of dynamical systems for statistical physics models out of equilibrium; and (ii) Study of exceptional points on the sample path of stochastic processes such as Brownian motion, random walks and stable processes. Emphasis is put on points defined by means of the corresponding occupation measures of sets of shrinking diameters. Other directions of proposed research involving these ingredients are applicable among other things to universal lossy coding, a problem of relevance and much interest for communication theory and to biomolecular data analysis. The theory of large deviations is mainly concerned with rare events or tail estimates on an exponential scale. This theory has proven to be successful as a tool for deriving almost sure asymptotic limits when two levels of randomness are present. Of particular interest are problems in which large deviations estimates and ideas play a decisive role in determining a limit law that holds with probability one.