Mathematical Sciences: Probability Measures on Vector Spaces: Basic Results and Application

本研究属于概率论领域，涉及实值和向量值随机变量的极限定理，重点是迭代对数定律（LIL）和高斯中心极限定理（CLT）。特别地，我们将研究由修剪和自归一化驱动的CLT和LIL的相关问题。高斯随机向量的二阶行为和过程的随机样本的极限集，以及覆盖问题，将被检查。其他正在考虑的项目涉及向量值部分和，经验过程，增长率，局部渐近性，以及Banach空间值布朗运动的逃逸率。概率论的中心主题，以及它在工程、物理和统计学问题中的各种应用，都是研究有限维和无限维随机变量部分和的极限定理。最重要的是强大数定律（SLLN），中心极限定理（CLT），以及SLLN的收敛速度，即迭代对数定律（LIL）。本课题将集中研究这些极限定理。

This research is in the area of probability and involves the limit theorems for real and vector valued random variables with emphasis on the law of the iterated logarithm (LIL) and the Gaussian central limit theorem (CLT). In particular, problems related to the CLT and LIL motivated by trimming and self- normalizations will be studied. The second order behavior of Gaussian random vectors and limit sets for random samples of processes, as well as coverage problems, will be examined. Other items under consideration concern vector valued partial sums, empirical processes, and the rates of growth, local asymptotics, and rates of escape for Banach space valued Brownian motion. The central theme in probability, and throughout its diverse applications to problems in engineering, physics, and statistics, is the study of limit theorems for partial sums of random variables in both finite and infinite dimensional settings. Of primary importance is the strong law of large numbers (SLLN), the central limit theorem (CLT), and the rate of convergence for the SLLN, namely the law of the iterated logarithm (LIL). This research project will be concentrated on the study of these limit theorems.