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

建议的研究中心随机不等式， 包括相关不等式，单调性的某些 关于分布参数的期望， 量化依赖结构的不等式，以及相关 多变量总体积极性的概念。 重点是 多变量分布和过程。 该提案还 是指在顺序分析中出现的相关概念， 不同条件下最优停止值比较的研究 依赖结构和预言不平等， 最优停止值的上界。 的极限定理 将研究各种类型的依赖结构， 最初强调近似率。 结果改善 收敛速度可以应用于渐近性的研究 可以表示为以下各项之和的多变量统计 相关随机变量和各种随机计数， 组合性质的问题 样本的统计独立性假设 测量是大多数数据分析方法的基础， 在随机优化问题中无处不在。 不过有一 在许多情况下，这一假设被违反。 拟议的研究集中在某些方面的研究， 样本由非独立的观测值组成。

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.