Limit Theorems in Probability Theory

0070382 Gine的工作计划在概率和统计中的渐近理论的几个主题上。 研究的一个主要目的是通过研究指数不等式和矩不等式（Rosenthal Pinelis不等式和伯恩斯坦不等式的$U$-统计量的真正类似物是什么？是否存在一致的伯恩斯坦不等式或一致的Prohorov不等式，例如最近的Talagrand不等式，用于独立随机变量和的集合？）和极限定理，特别是重对数定律。 这些结果可能会得到广义U$-统计量，包括独立随机变量的多线性形式。 这些主题在统计学中的应用，特别是审查数据，也将继续进行。 研究的第二个对象是独立随机变量的自归一化和，特别是与自助法和学生t-统计量。 最后私家侦探也有兴趣探索经验过程的现代理论及其技术在不同领域的应用，如多粒子系统的占领测量过程的波动的渐近性，以及基于经验过程的不同泛函的估计和测试。经验测量是对一系列数据点描述的简写。 独立随机变量和经验过程的和可以被认为是一个变量函数关于这个测度的单积分，而$U$-统计量和$U$-过程可以被认为是多个变量函数关于经验测度的多重积分。 一阶渐近统计通常基于独立随机变量和过程的极限定理，但更精确的二阶性质需要U-统计和过程的极限理论（在某种程度上，类似于在微积分中研究函数时使用高阶导数而不是仅使用一阶导数）。 虽然U统计量是在40年代引入的，但直到最近，它们的渐近理论才接近于最终形式，部分原因是这位物理学家以前的努力。合作者;所提出的研究目的是通过获得最佳可能的分布不等式、矩不等式和重对数律来完成U-统计的古典概率这一章，并推进U-过程的理论。 这项研究还将包括生存分析中的应用。在另一个方向上，人们普遍认为，用依赖于自身的某些量而不是数值常数来规范独立随机变量的和可以改善收敛性质（特别是，基于这种自规范化量的统计过程可能具有良好的性质，这方面最主要和最古老的例子是著名的学生t统计量和检验）。 但这一点必须在每一个实例中表现出来。 私家侦探我想研究一些与自正规化和有关的问题，特别是与自助法有关的问题。 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）。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），还有私家侦探我想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.