Mathematical Sciences: Tail Probability Approximations for Sums of Dependent Variables

本文研究的问题是如何准确地描述相关随机变量和的概率行为。拟议工作的基本工具将是解耦不平等领域的最新成果。解耦不等式将相关随机变量和问题转化为条件独立随机变量和问题。在建议的工作中，特别注意将集中在发展随机停止和的尾部概率近似，独立随机变量的二次形式，以及任意依赖随机变量和的一般问题。本研究旨在探讨概率论中的一些重要问题。特别地，研究者将观察依赖观察的总和的行为。这些总和将通过与独立观测的适当总和进行比较来研究。

This research is to study the problem of obtaining an accurate description of the probabilistic behavior of sums of dependent random variables. Essential tools for the proposed work will be recent results in the area of decoupling inequalities. A decoupling inequality transforms problems on sums of dependent random variables to problems on sums of conditionally independent random variables. In the proposed work, particular attention will be focused on developing tail probability approximations for randomly stopped sums, quadratic forms of independent random variables, and the general problem of sums of arbitrarily dependent random variables. This research is to study some important problems in probability theory. In particular, the investigator will look at the behavior of sums of dependent observations. These sums will be studied by comparing them to appropriate sums of independent observations.