Limit Theorems and Inequalities in Probability

9626778 Zinn摘要这位研究者致力于研究关于U-统计和U-过程的结构问题，对称凸集的相关不等式和函数的超压缩性（除了和）。U-统计和过程项目是对本研究员J. Cuzick和E. Gine以及Anda Gadidov（刚刚获得博士学位）的一些工作。在调查员的指导下）。关于相关不平等问题的工作项目已经进行了一段时间，并已取得若干成果。特别是，Schechtman，Schlumprecht和这个调查员获得了所需的不等式椭球。 第三个项目跟进最近提交的论文（与Hitczenko，Kwapien，Li，Schechtman，Schlumprecht）关于多指标独立，同分布随机变量交替最大值和最小值的超收缩性。 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来获得关于“人口”基本特征的信息。现代理论，在过去二十年左右的发展，允许一个研究许多这样的特点，同时一个组成部分的古典和现代理论研究本调查员是U-过程，这是，在某种意义上说，积木一般统计泛函。 这种调查将有助于更好地评估目前用于医学、天文学和其他领域审查和/或截断数据研究的统计程序的准确性。 经验过程的现代理论也与概率的另一个领域，即高斯过程的研究相联系。不等式的过程中发挥了重要作用，不仅在概率-以及许多其他领域的数学，如调和分析，巴拿赫空间理论，最近，运营商理论-而且在理论物理（量子场论）。 高斯相关猜想就是这一领域中的一个突出问题。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障碍问题。人们希望解决这个问题的副产品是解决各个领域存在的其他双边问题。解决这个问题的方法之一，导致了本研究员正在研究的第三个项目，即收缩过度。在我们的上下文中hypercontractivity允许一个改变原来的制定的问题，一个问题更适合一些最近的进展，在理论的不平等高斯过程。

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..