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

9626175 de la Pena摘要对相依随机变量和的行为的研究在概率和统计的理论和应用中都起着核心作用。U-统计量在与估计有关的问题中经常遇到，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的可逆性有关的研究中。独立随机变量的随机停止和是序贯分析研究的核心，也是排队论、库存论、可靠性理论等领域的研究热点。这位研究人员和他的同事们考虑了涉及相依随机变量和的相当一般的问题，包括Wald方程在鞅、二次型和双重随机积分情况下的推广；多线性形式的尾概率的近似；条件化原理对更广泛类别问题的推广；以及相关极限定理的收敛速度的确定。在解决提出的问题时，这位研究人员和他的同事们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。这位研究人员和他的同事们处理了几个问题，这些问题的解决将对概率和统计的几个领域产生有益的影响。这一广泛的领域包括对表现出高度相互依存的现象的性质的研究，因此很难单独分析。困难源于与其他组件之间存在的紧密联系。研究人员和他的同事在处理这类问题时所遵循的方法是引入一组新的现象，这些现象与原来的现象非常相似，但除此之外还具有理想的独立性。在概率语言中，人们将这种处理依赖的方法称为现象依赖的解耦。到目前为止，在这一领域已经取得了几项成果，这些成果通常会产生最佳结果。这位研究人员和他的同事们继续将这一理论应用于序贯分析、U统计和随机积分中的问题。序列分析的统计学理论在第二次世界大战期间被引入，作为优化资源的一种手段。与分配前置样本并仅在获得样本量后才分析数据的典型统计方法形成鲜明对比的是，序贯方法允许通过在实验的每个阶段密切跟踪过程的发展来优化资源。序贯方法在破坏性抽样的情况下特别有用，例如在军事装备和工业应用中的设备和补给寿命研究中。在医学研究中，它的应用既可以优化资源，又可以处理在有强烈迹象表明研究中的药物有害或被证明有益时提前终止临床试验的伦理问题，而不必等待按照经典方法获得预先确定的样本量。

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.