Mathematical Sciences: Limit Theorems and Inequalities in Probability

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The principal investigator will work on several projects. The first focusses on extremal problems involving (generalized) moment inequalities for sums of independent, symmetric random variables with an eye toward obtaining best constants. The research builds on the classic inequalities of Khintchine and Rosenthal. The from of the final inequalities should give some insight into the nature of cancellation phenomena and may suggest extensions to more general dependence relations. This work is ongoing and is joint work with T. Figiel, P. Hitczenko and W. B. Johnson. The second project concerns structural problems for U- statistics and U-processes. Gine and the principal investigator have found very little information concerning necessity in the context of U-statistics. Together with J. Cuzik, they have made some progress on a very special case, but there is much more work required to understand the general question of cancellation in this context. Two other projects deal with trying to find usable necessary and sufficient conditions for a class of functions to be uniform Donsker (with E. Gine) and studying questions concerning convergence rates in the context of bootstrapping empirical measures (with F. Gotze). The principal investigator is working on several projects whose primary focus is a better understanding of cancellation phenomena for sums of independent random variables. These types of problems often arise in statistics.