Limit Theorems and Inequalities in Probability

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9626778 Zinn ABSTRACT This investigator works on structural problems concerning U-statistics and U-processes, correlation inequalities for symmetric convex sets and hypercontractivity for functions (other than sums). The project on U-statistics and processes follows up on fairly recent work of this investigator, J. Cuzick and E. Gine as well as some work of Anda Gadidov (who has just obtained her Ph.D. under the direction of this investigator). The project on correlation inequalities work has been under way for some time and several results have already been obtained. In particular, Schechtman, Schlumprecht and this investigator obtained the desired inequality for ellipsoids. The third project follows up on a recently submitted paper (with Hitczenko, Kwapien, Li, Schechtman, Schlumprecht) on hypercontractivity of alternating maxima and minima of multi-indexed independent, identically distributed random variables. Classical empirical processes can and have been used by statisticians to obtain information about basic characteristicq of a ``population'' using samples collected from the population. The modern theory, developed over the last twenty or so years, allows one to study many such characteristics simultaneously A component of both the classical and modern theory studied by this investigator is U-processes, which are, in a sense, building blocks for general statistical functionals. Such investigations will help to better assess the accuracy of statistical procedures currently used in the study of censored and/or truncated data in Medicine, Astronomy and other fields. The modern theory of empirical processes has also made connections with another field of probability, namely, the study of Gaussian processes. Inequalities for such processes have played a major role not only in probability--as well as many other fields of mathematics, such as harmonic analysis, Banach space theory and, more recently, operator theory--but also in Theoretical Physics (Quant um field theory). The Gaussian Correlation Conjecture is one such outstanding problem in this area. Many correlation results in, for example, statistical mechanics, attack one-sided problems. This particular conjecture attacks a two-sided barrier problem. One hoped for by-product of a solution to this problem is to other two-sided problems present in a variety of fields. One of the approaches to this problem, led to the third project under study by this investigator, namely, hypercontractivity. In our context hypercontractivity allows one to change the original formulation of the problem, to a problem more amenable to some recent advances in the theory of inequalities for Gaussian processes..