Mathematical Sciences: Limit Theorems and Inequalities in Probability

首席研究员将从事几个项目。第一部分集中在涉及（广义）独立对称随机变量和的矩不等式的极端问题上，着眼于获得最佳常数。这项研究建立在Khintchine和Rosenthal的经典不等式基础上。最后不等式的推导应该对消去现象的性质有一些深入的了解，并可能提出对更一般的依赖关系的扩展。这项工作正在进行中，由T. Figiel、P. Hitczenko和W. B. Johnson共同完成。第二个项目涉及U型统计和U型过程的结构性问题。Gine和首席研究员在u型统计的背景下发现了很少关于必要性的信息。与J. Cuzik一起，他们在一个非常特殊的情况下取得了一些进展，但要理解这种情况下的一般取消问题，还需要做更多的工作。另外两个项目涉及试图找到一类函数是一致Donsker的可用的必要和充分条件（与E. Gine一起）和研究关于自举经验测度背景下的收敛率问题（与F. Gotze一起）。首席研究员正在进行几个项目，其主要重点是更好地理解独立随机变量和的消去现象。这类问题经常出现在统计学中。

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.