Limit Theorems in Probability Theory

University Of Connecticut, Storrs CT

Investigators

Abstract

0070382 Gine Work is planned on several topics from asymptotic theory in Probability and Statistics. A main thrust of the research aims at deepening our understanding of canonical $U$-statistics and $U$-processes by investigating exponential and moment inequalities (what are the true analogues for $U$-statistics of the Rosenthal-Pinelis and Bernstein's inequalities? is there a uniform Bernstein, or uniform Prohorov inequality such as the recent inequality of Talagrand for collections of sums of independent random variables?) and limit theorems, particularly the law of the iterated logarithm. These results may be obtained for generalized $U$-statistics, including multilinear forms in independent random variables. Applications of these topics in Statistics, particularly censored data, will also be pursued. A second object of study are selfnormalized sums of independent random variables, particularly in connection with the bootstrap and with the Student t-statistic. Finally, the P.I. is also interested in exploring the application of the modern theory of empirical processes and its techniques in different areas such as asymptotics of the fluctuations of the occupation measure process for multiple particle systems, and estimation and testing based on different functionals of the empirical process. The empirical measure is shorthand for the description of a series of data points. Sums of independent random variables and empirical processes can be thought of as single integrals of functions of one variable with respect to this measure, and $U$-statistics and $U$-processes, as multiple integrals with respect to the empirical measure of functions of several variables. First order asymptotic statistics is often based on limit theorems for sums of independent random variables and processes, but more refined second order properties require limit theory for $U$-statistics and processes (in a way, in analogy with the use of higher order derivatives versus only the first derivative when studying functions in Calculus). Although $U$-statistics were introduced in the forties, their asymptotic theory has not been close to reaching its final form until recently, in part due to previous efforts by this P.I. and collaborators; the proposed research aims at completing this chapter of Classical Probability for $U$-statistics, and at advancing the theory of $U$-processes, by obtaining best possible distributional and moment inequalities and laws of the iterated logarithm. This research will also include applications in survival analysis. In another direction, it is accepted wisdom that normalizing sums of independent random variables by certain quantities that depend on themselves rather than numerical constants improves the convergence properties (in particular, then, statistical procedures based on such selfnormalized quantities may have good properties, the leading and oldest example of this being the famous Student t-statistic and test). But this must be shown at each instance. The P.I. would like to study some questions related to selfnormalized sums, particularly in connection with the bootstrap. Empirical process theory vigorously developed during the last two decades (with substantial contributions by this P.I.) and, since then, its impact on different fields of stochastics has not ceased to increase (in classical asymptotic statistics, information theory, neural networks, machine learning, model selection, statistical mechanics, etc.), and the P.I. would like to continue applying it to different statistics and probability problems of current interest.