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摘要研究相关随机变量总和的行为在概率和统计的理论和应用中都起着核心作用。在有关估计的问题中通常会遇到U统计量，而在与多个随机整合，回归和协方差分析以及矩阵的可逆性有关的研究中出现了多线性形式。在顺序分析研究的核心以及排队理论，库存理论和可靠性理论等不同领域的核心发现了随机停止独立随机变量和martingles的总和。研究者及其同事认为涉及依赖随机变量总和的相当普遍的问题，包括在Martingales，二次形式和双重随机积分的情况下扩展WALD方程；多线性形式的尾巴概率的近似；将调节原则的概括为更广泛的问题；并确定相关限制定理的收敛速度。在解决拟议问题的过程中，研究人员及其同事从他们最近的结果中汲取灵感，其中包括对顺序分析，U统计和经验过程的理论的两个基本贡献。 研究人员及其同事处理了一些问题，这些问题将对几个概率和统计领域产生有益的影响。广泛的领域包括对表现出高水平相互依存的现象的性质的研究，因此很难自行分析。困难源于与其他组件的牢固联系。在处理这种类型的问题时，调查员及其同事的方法在于引入一组新现象，这些现象与原始的现象非常相似，但此外还具有理想的独立性。用概率语言，有人将这种方法称为处理依赖性的依赖性的依赖性。到目前为止，该领域有几个结果通常会产生最佳结果。研究者及其同事继续将该理论应用于顺序分析，U统计和随机整合的问题。顺序分析的统计理论是在第二次世界大战期间引入的，作为优化资源的一种手段。与典型的统计方法形成鲜明对比的是，分配前缀样品并仅在实现样本量后才分析数据的典型方法，顺序方法可以通过在实验的每个阶段紧密遵循过程的开发来优化资源。顺序方法在破坏性采样的情况下特别有用，就像在军事设备和工业应用中的设备和终生研究中一样。在医学研究中，当有很大的迹象表明所研究的药物有害或证明是有益的，而不必等到根据经典方法获得预固定的样本量，它的应用既可以优化资源，又可以处理终止临床试验的道德问题。