Limit Theorems in Probability Theory

0070382从概率统计中的渐近理论出发，计划在几个主题上开展工作。这项研究的一个主要目的是通过调查指数和矩不等(罗森塔尔-皮内利斯和伯恩斯坦不等式的统计)来加深我们对规范的$U$-统计和$U$-过程的理解($U$的真正类似是什么？对于独立随机变量和的集合，是否存在一致的Bernstein，或一致的Prohorov不等式，如最近的TALAGRAND不等式？)极限定理，特别是重对数律。这些结果可用于广义$U$-统计量，包括独立随机变量中的多线性形式。还将探讨这些主题在统计学中的应用，特别是经审查的数据。第二个研究对象是独立随机变量的自归一化和，特别是与自举和学生t统计量有关的。最后，P.I.还有兴趣探索现代经验过程理论及其技术在不同领域的应用，如多粒子系统占领测量过程波动的渐近性，以及基于经验过程不同泛函的估计和检验。经验度量是对一系列数据点的描述的缩写。独立随机变量与经验过程之和可以看作是一个变量函数关于这一测度的单积分，而$U$-统计量和$U$-过程可以看作是关于多个变量函数的经验测度的多重积分。一阶渐近统计量通常基于独立随机变量和过程的极限定理，但更精细的二阶性质需要$U$统计量和过程的极限理论(在某种程度上，类似于在研究微积分中的函数时使用更高阶导数而不是只使用一阶导数)。虽然$U$-统计是在四十年代引入的，但他们的渐近理论直到最近才接近最终形式，部分原因是这位P.I.和合作者以前的努力；拟议的研究旨在完成$U$-统计的经典概率这一章，并通过获得最佳可能的分布和矩不等以及重对数律来推进$U$-过程理论。这项研究还将包括在生存分析中的应用。另一方面，人们普遍认为，将独立随机变量的和归一化为依赖于它们自身而不是数字常量的某些量可以改善收敛性质(特别是，基于这种自归一化量的统计过程可能具有良好的性质，这方面的主要和最古老的例子是著名的学生t统计和检验)。但这必须在每一个实例中显示出来。P.I.想要研究一些与自归一化和有关的问题，特别是与自举相关的问题。经验过程理论在过去的二十年里蓬勃发展(这位P.I.做出了重大贡献)。而且，从那时起，它对随机学不同领域的影响一直在增加(在经典渐近统计、信息论、神经网络、机器学习、模型选择、统计力学等方面)，P.I.希望继续将其应用于当前感兴趣的不同统计和概率问题。

0070382Gine Work is planned on several topics from asymptotic theory in Probability and Statistics. A main thrust of the research aims at deepening our understanding of canonical $U$-statistics and $U$-processes by investigating exponential and moment inequalities (what are the true analogues for $U$-statistics of the Rosenthal-Pinelis and Bernstein's inequalities? is there a uniform Bernstein, or uniform Prohorov inequality such as the recent inequality of Talagrand for collections of sums of independent random variables?) and limit theorems, particularly the law of the iterated logarithm. These results may be obtained for generalized $U$-statistics, including multilinear forms in independent random variables. Applications of these topics in Statistics, particularly censored data, will also be pursued. A second object of study are selfnormalized sums of independent random variables, particularly in connection with the bootstrap and with the Student t-statistic. Finally, the P.I. is also interested in exploring the application of the modern theory of empirical processes and its techniques in different areas such as asymptotics of the fluctuations of the occupation measure process for multiple particle systems, and estimation and testing based on different functionals of the empirical process. The empirical measure is shorthand for the description of a series of data points. Sums of independent random variables and empirical processes can be thought of as single integrals of functions of one variable with respect to this measure, and $U$-statistics and $U$-processes, as multiple integrals with respect to the empirical measure of functions of several variables. First order asymptotic statistics is often based on limit theorems for sums of independent random variables and processes, but more refined second order properties require limit theory for $U$-statistics and processes (in a way, in analogy with the use of higher order derivatives versus only the first derivative when studying functions in Calculus). Although $U$-statistics were introduced in the forties, their asymptotic theory has not been close to reaching its final form until recently, in part due to previous efforts by this P.I. and collaborators; the proposed research aims at completing this chapter of Classical Probability for $U$-statistics, and at advancing the theory of $U$-processes, by obtaining best possible distributional and moment inequalities and laws of the iterated logarithm. This research will also include applications in survival analysis. In another direction, it is accepted wisdom that normalizing sums of independent random variables by certain quantities that depend on themselves rather than numerical constants improves the convergence properties (in particular, then, statistical procedures based on such selfnormalized quantities may have good properties, the leading and oldest example of this being the famous Student t-statistic and test). But this must be shown at each instance. The P.I. would like to study some questions related to selfnormalized sums, particularly in connection with the bootstrap. Empirical process theory vigorously developed during the last two decades (with substantial contributions by this P.I.) and, since then, its impact on different fields of stochastics has not ceased to increase (in classical asymptotic statistics, information theory, neural networks, machine learning, model selection, statistical mechanics, etc.), and the P.I. would like to continue applying it to different statistics and probability problems of current interest.