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

摘要本文主要研究出现在以下几种情况下的随机不等式和依赖结构：a)正相关检验统计量的多重检验，目的是用适当相关检验的有用界代替独立检验的已知错误概率率。b)因然随机变量的先知不等式，它本质上提供了对顺序观测数据中未来信息价值的度量。c)正负相关不等式：从随机图上的某些分布得到的关于随机变量的相关不等式和负相关不等式的一些猜想被证明等价于有趣的行列式不等式。特殊情况已直接证明，并通过数值计算。另一个猜想与积分不等式有关，它扩展了著名的van Den Berg-Kesten猜想（传闻最近已被证明），并提出了有趣的新应用。利用FKG型不等式和重排不等式建立了新猜想的特殊情况；数值计算似乎支持这些扩展。d)因缘随机变量的极限定理，目的是利用Stein的方法和经典方法，根据相关结构的有意义的特征获得收敛速率。抽样测量的统计独立性假设是许多经典数据分析的基础。当人们可以假设数据来自重复的对照实验，其中重复之间没有误差的延续时，这是相关的。然而，具有更复杂数据的情况是普遍存在的。建议的研究重点是研究由非独立的观察（随机变量）组成的样本的某些方面，以及依赖结构的各种模型。这种结构也与物理中的各种模型有关，并且还研究了例如渗流理论中出现的某些问题。本研究的另一个方面与统计大样本理论有关。统计理论的很大一部分是建立在只对大样本有效的近似基础上的。研究了相关数据的近似方法，以确定该理论在复杂情况下的有效性。在决定多大的样本才能使大样本近似有效时，近似的质量很重要。

9504614 Rinott Abstract The proposed research centers on stochastic inequalities and dependence structures which arise in several contexts: a) Multiple tests for positively dependent test statistics, with the goal of replacing known error probability rates for independent tests by useful bounds for suitably dependent tests. b) Prophet inequalities for dependent random variables, which essentially provide measures on the value of future information in sequentially observed data. c) Positive and negative dependence inequalities: some conjectures on correlation inequalities and negative dependence inequalities for random variables obtained from certain distributions on random graphs are shown to be equivalent to interesting determinental inequalities. Special cases have been proved directly, and by numerical calculations. Another conjecture concerns integral inequalities which extend the well-known van Den Berg-Kesten Conjecture (rumored to have been proved recently) and suggest interesting new applications. Special cases of the new aonjecture have already been established using FKG type and rearrangement inequalities; numerical calculations seem to support these extensions. d) Limit theorems for dependent random variables, with the goal of obtaining convergence rates in terms of meaningful characteristics of the dependence structures, using Stein's method and classical approaches. The assumption of statistical independence of sampled measurements is basic to much of classical data analysis. It is relevant when one can assume that the data arises from repeated controlled experiments in which there is no carryover of errors between repetitions. However, situations with more complex data are ubiquitous. The proposed research centers on the study of certain aspects of samples which consist of observations (random variables) which are not independent, and various models for dependence structures. Such structures are also relevant in various models in physics, and certain issues arising in percolation theory, for example, are also studied. Another aspect of this research is relevant to statistical large sample theory. A good portion of statistical theory is based on approximations which are valid only for large samples. Methods for such approximations for dependent data are investigated, in order to determine the validity of the theory in complex situation. The quality of the approximations is important in deciding how large samples must be in order for large sample approximations to be valid.