Limit Theorems and Inequalities in Probability

9626778 Zinn摘要这位研究者致力于U-统计量和U-过程、对称凸集的相关不等式和函数(和除外)的超紧性等结构性问题的研究。关于U-统计和过程的项目是对这位研究员J.Cuzick和E.Gine最近的工作以及Anda Gadidov(她刚刚在这位研究员的指导下获得博士学位)的一些工作的后续行动。关于相关不平等的项目已经开展了一段时间，并且已经取得了一些成果。特别地，谢克特曼、斯伦普雷希特和这位研究者得到了椭球上所期望的不等式。第三个项目是对最近提交的一篇论文(与Hitczenko，Kuapien，Li，Scheck htman，SchLumprecht)关于多指标独立、同分布随机变量的交替极大值和极小值的超紧性的后续研究。统计学家可以而且已经使用经典的经验过程，利用从人口中收集的样本，获得有关“人口”的基本特征的信息。在过去二十年左右的时间里发展起来的现代理论允许人们同时研究许多这样的特征。这位研究者研究的古典和现代理论的一个组成部分是U-过程，在某种意义上，它是一般统计泛函的基石。这类调查将有助于更好地评估目前在医学、天文学和其他领域的审查和/或截断数据研究中使用的统计程序的准确性。现代经验过程理论也与另一个概率领域联系在一起，即对高斯过程的研究。这类过程的不等式不仅在概率学中发挥了重要作用--也在许多其他数学领域，如调和分析、Banach空间理论和最近的算子理论--而且在理论物理(量子场论)中也发挥了重要作用。高斯关联猜想就是这一领域的一个突出问题。例如，在统计力学中，许多相关性导致了片面的问题。这个特殊的猜想解决了一个双边障碍问题。解决这一问题的一个希望副产品是存在于各种领域的其他两面性问题。解决这一问题的方法之一，导致了这位研究者正在研究的第三个项目，即过度收缩。在我们的上下文中，超收缩允许一个人改变问题的原始形式，变成一个更适合于高斯过程的不等式理论的一些最新进展的问题。

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..