Limit Theorems and Inequalities in Probability

极限定理和概率不等式

• 批准号: 9626778
• 负责人: Joel Zinn
• 金额: $ 3.9万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626778&HistoricalAwards=false
• 关键词: Limit; Theorems; Inequalities; Probability

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..

摘要本研究者研究关于u统计量和u过程的结构问题，对称凸集的相关不等式和函数（除和外）的超收缩性。关于u型统计和过程的项目是在这位研究者J. Cuzick和E. Gine最近的工作，以及安达·加迪多夫（她刚刚在这位研究者的指导下获得博士学位）的一些工作的基础上进行的。有关不等式工作的项目已经进行了一段时间，已经取得了一些成果。特别地，Schechtman， Schlumprecht和这个研究者得到了椭球体所需的不等式。第三个项目跟进了最近提交的一篇论文（与Hitczenko， Kwapien, Li, Schechtman, Schlumprecht）关于多索引独立同分布随机变量的交替极大值和极小值的超收缩性。统计学家可以并且已经使用经典的经验过程，通过从总体中收集样本来获得关于“总体”基本特征的信息。在过去20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。这位研究者研究的经典理论和现代理论的一个组成部分是u过程，它在某种意义上是一般统计泛函的组成部分。这种调查将有助于更好地评估目前用于研究医学、天文学和其他领域的删减和（或）删减数据的统计程序的准确性。经验过程的现代理论也与另一个概率论领域，即高斯过程的研究，建立了联系。这些过程的不等式不仅在概率论以及许多其他数学领域（如谐波分析、巴拿赫空间理论和最近的算子理论）中发挥了重要作用，而且在理论物理学（量子场论）中也发挥了重要作用。高斯相关猜想就是这一领域的突出问题之一。许多相关的结果，例如，统计力学，攻击片面的问题。这个特殊的猜想攻击了一个双边势垒问题。一个希望解决这个问题的副产品是解决在各种领域中存在的其他双边问题。解决这个问题的方法之一，导致了这位研究者正在研究的第三个项目，即过度收缩。在我们的背景下，超收缩性允许人们改变问题的原始表述，使问题更适合于高斯过程不等式理论的一些最新进展。