Mathematical Sciences: Lp and Tail Probability Approximations for Sums of Dependent Variables

数学科学：因变量和的 Lp 和尾部概率近似

• 批准号: 9626175
• 负责人: Victor de la Pena
• 金额: $ 7.43万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626175&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Lp; Tail; Probability

9626175 de la Pena ABSTRACT The study of the behavior of sums of dependent random variables plays a central role in both the theory and applications of probability and statistics. U-statistics are commonly encountered in problems concerning estimation, while multilinear forms arise in research pertaining to multiple stochastic integration, regression and covariance analysis, and invertibility of matrices. Randomly stopped sums of independent random variables and martingales are found at the core of the studies of sequential analysis, and such diverse areas as queuing theory, inventory theory and reliability theory. The investigator and his colleagues consider fairly general problems involving sums of dependent random variables, including extensions of Wald's equation in the case of martingales, quadratic forms and double stochastic integrals; approximations of the tail probabilities of multilinear forms; generalization of the principle of conditioning to a wider class of problems; and determination of speeds of convergence of related limit theorems. In tackling the proposed problems, the investigator and his colleagues draw from their recent results which include two fundamental contributions to the theories of sequential analysis, U-statistics and empirical processes. The investigator and his colleagues deal with several problems whose solution would have a beneficial impact on several areas of probability and statistics. The broad area consists of the study of the properties of phenomena that exhibit high levels of inter-dependence and, hence, are hard to analyze on their own. The difficulty stems from the strong links present with other components. The approach followed by the investigator and his colleagues when dealing with this type of problem consists in introducing a new set of phenomena that closely resembles the original one but which in addition has desirable independence properties. In probabilistic language, one calls this approach to dealing with dependence a de-coupling of t he dependence of a phenomenon. Up to now there are several results developed in this area which typically produce optimal results. The investigator and his colleagues continue to apply this theory to problems in sequential analysis, U-statistics and stochastic integration. The statistical theory of sequential analysis was introduced during World War II as a means of optimizing resources. In sharp contrast with the typical statistical approach of assigning a prefixed sample and analyzing the data only after the sample size has been achieved, the sequential approach permits the optimization of resources by closely following the development of the process at each stage of the experiment. The sequential approach is particularly useful in cases of destructive sampling, as in equipment and supply lifetime studies in military equipment and industrial applications. In medical studies, its application allows both to optimize resources and to deal with the ethical issues of terminating early a clinical trial when there is strong indication that the drug under study is harmful or proven to be beneficial, without having to wait until a pre-fixed sample size has been attained in accordance with the classical approach.

9626175 de la Pena 摘要 对因随机变量之和的行为的研究在概率和统计的理论和应用中发挥着核心作用。 U 统计量通常在估计问题中遇到，而多线性形式则出现在与多重随机积分、回归和协方差分析以及矩阵可逆相关的研究中。独立随机变量和鞅的随机停止总和是序贯分析以及排队论、库存理论和可靠性理论等不同领域研究的核心。研究人员和他的同事考虑了涉及相关随机变量之和的相当普遍的问题，包括在鞅、二次形式和双随机积分情况下 Wald 方程的扩展；多线性形式尾部概率的近似；将调节原理推广到更广泛的问题；以及确定相关极限定理的收敛速度。在解决所提出的问题时，研究者和他的同事借鉴了他们最近的结果，其中包括对序贯分析理论、U 统计和经验过程的两个基本贡献。 研究人员和他的同事处理了几个问题，这些问题的解决将对概率和统计的多个领域产生有益的影响。这个广泛的领域包括对表现出高度相互依赖的现象的属性的研究，因此很难单独分析。困难源于与其他组件的紧密联系。研究者和他的同事在处理此类问题时采用的方法包括引入一组新的现象，这些现象与原始现象非常相似，但另外还具有理想的独立性。用概率语言来说，人们将这种处理依赖性的方法称为现象依赖性的解耦。到目前为止，该领域已经取得了一些成果，通常会产生最佳结果。研究人员和他的同事继续将该理论应用于序贯分析、U 统计和随机积分中的问题。序贯分析的统计理论是在第二次世界大战期间引入的，作为优化资源的手段。与分配前缀样本并仅在达到样本量后才分析数据的典型统计方法形成鲜明对比，顺序方法允许通过密切跟踪实验每个阶段的过程的发展来优化资源。顺序方法在破坏性采样的情况下特别有用，例如军事设备和工业应用中的设备和电源寿命研究。在医学研究中，它的应用既可以优化资源，也可以在有强烈迹象表明所研究的药物有害或被证明有益时处理提前终止临床试验的伦理问题，而不必等到预审阶段。根据经典方法获得了固定样本量。