Mathematical Sciences: General Limit Theorems in Probability and Local Empirical Processes

9503665 Einmahl摘要这个建议解决了三个不同领域的问题：(1)关于独立随机变量和的行为的普遍结果(2)局部经验过程(3)Banach空间中的一般重对数律。区域(1)的工作是由最近关于独立随机变量和的几乎必然行为的一些普遍结果所推动的。与这个方向的经典结果相反，这些结果不需要矩类型条件。主要的困难是找到合适的正规化和居中序列，这些序列特定于所调查的特定情况。在其他方面，将使用分位数转换。区域(2)的工作主要致力于局部经验过程的极限行为，这些过程在密度估计和回归函数估计中有很好的应用。这一建议的第三部分最后解决了与Banach空间中最近的一个一般重对数律有关的一些问题。这一建议解决了概率和经验过程中关于极限定理的一些问题。概率中的极限定理，如中心极限定理和大数定律，为许多广泛使用的处理大型数据集的统计技术提供了基础。然而，这些结果中的一些只能在相对限制性的条件下应用。在过去的十年里，在概率方面发展了强大的方法。人们建议使用这些新技术来进一步扩展基本极限定理的适用性，同时，对经典技术失败的情况有一些了解。还将考虑统计估计理论的一些应用，其中主要重点将是所谓的经验过程。

9503665 Einmahl Abstract This proposal addresses questions from three different areas: (1) Universal Results on the Behavior of Sums of Independent Random Variables (2) Local Empirical Processes (3) A General Law of the Iterated Logarithm in Banach space. The work in area (1) is motivated by some recent universal results on the almost sure behavior of sums of independent random variables. Contrary to the classical results in this direction, these results do not require moment type conditions. The main difficulty is to find suitable norming and centering sequences which are specific to the particular situations under investigation. Among other things, quantile transformations will be used. The work from area (2) is mainly devoted to the limit behavior of local empirical processes which have useful applications in density and regression function estimation. The third part of this proposal finally addresses a number of questions in connection with a recent general law of the iterated logarithm in Banach space. This proposal addresses a number of questions on limit theorems in probability and empirical processes. Limit theorems in probability such as the central limit theorem, and laws of large numbers provide the basis for many widely used statistical techniques to handle large data sets. Some of these results, however, can only be applied under relatively restrictive conditions. Over the last decade, powerful methods have been developed in probability. It is proposed to use these new techniques to further extend the applicability of the basic limit theorems, and, at the same time, to gain some insight into situations, where the classical techniques have failed. Some applications to statistical estimation theory will be considered as well, where the main emphasis will be on so-called empirical processes.