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

这项研究是在概率领域，涉及 随机变量的真实的和向量值极限定理 强调迭代对数律（LIL）和 高斯中心极限定理（CLT）。 特别是， 与CLT和LIL有关， 规范化将被研究。 的二阶行为 随机样本的高斯随机向量和极限集 将审查程序以及覆盖问题。 其他 所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， Banach空间值布朗运动的逃逸率。 概率的中心主题，以及其多样的 应用于工程、物理和统计问题， 是研究随机变量部分和的极限定理 在有限和无限维设置变量。 的 强大数定律（Strong Law of Large Numbers，SLLN） 中心极限定理（CLT），和收敛速度的 SLLN，即重对数律（LIL）。 这 研究项目将集中在这些限制的研究 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.