Mathematical Sciences: Stochastic Inequalities, Dependence Structures, and Some Associated Limit Theorems

建议的研究重点是随机不等式，包括相关不等式，分布参数的某些期望的单调性，量化依赖结构的不等式，以及多元总正性的相关概念。重点是多变量分布和过程。本文还参考了序列分析和各种依赖结构下最优停止值比较研究中出现的依赖概念，以及提供最优停止值上界的先知不等式。各种类型的依赖结构的极限定理将被研究，最初的重点是近似率。由此得到的改进的收敛率可适用于多元统计量的渐近性研究，多元统计量可以表示为组合性质问题中产生的相关随机变量和各种随机计数的和。抽样测量的统计独立性假设是大多数数据分析方法的基础，在随机优化问题中普遍存在。然而，在许多情况下，人们会遇到违反这一假设的情况。拟议的研究集中在研究样本的某些方面，这些方面由不独立的观察组成。

The proposed research centers on stochastic inequalities, including correlation inequalities, monotonicity of certain expectations with respect to parameters of distributions, inequalities which quantify dependence structures, and related notions of multivariate total positivity. The emphasis is on multivariate distributions and processes. The proposal also refers to dependence concepts arising in sequential analysis and the study of optimal stopping value comparisons under various dependence structures, and prophet inequalities which provide upper bounds on optimal stopping values. Limit theorems for various types of dependence structures will be studied, with initial emphasis on approximation rates. The resulting improved convergence rates may be applicable to the study of asymptotics of multivariate statistics which can be expressed as sums of dependent random variables and various random counts arising in problems of combinatorial nature. The assumptions of statistical independence of sampled measurements is basic to most methods of data analysis, and is ubiquitous in problems of stochastic optimization. However, one encounters many situations in which this assumption is violated. The proposed research centers on the study of certain aspects of samples which consist of observations which are not independent.