Processes With Dependent Increments: Boundary Crossing, Self-Normalization and Limit Theorems

首席研究员(PI)考虑涉及因变量和的问题，其解决方案需要使用解耦理论的现有结果和新结果。通过使用专门为此目的设计的不等式，解耦理论在将涉及因变量的问题转化为涉及自变量的问题方面非常成功。在这项研究中，PI将研究三个主要问题，并考虑可能用于解决这些问题的方法(包括解耦方法)。第一个问题涉及一种普遍而自然的方法，即根据随机过程的上确界的期望值所确定的函数来近似随机过程到达边界的预期等待时间。这种方法允许在开发有用的近似值时使用历史平均值。第二个问题涉及根据已知的自变量和的结果来逼近因变量和的收敛速度。这个问题有可能广泛地扩展条件原理的适用性，条件原理是一种推广了鞅中心极限定理的方法。第三个问题发展了自规格化鞅的概率近似。这个问题的结果有可能影响概率和统计学的几个领域，包括假设检验和可信区间理论，因为典型的自归一化变量在构建可信区间和检验假设时起着关键作用。在统计和概率方法的发展中，一个常见的假设是关于变量或过程是独立的或具有独立属性的断言，即其中一个变量发生的事情不会影响另一个变量发生的事情。这种独立性假设具有简化问题的数学复杂性的效果，并且在实践中相当常见。由于这种简化，在变量相互依赖的情况下，寻求允许使用涉及独立变量的结果的方法似乎是很自然的。国际和平研究所通过发展以上述范例为目标的“脱钩”理论，大力参与了这一项目。他在当前研究中的第一个问题是发展一种概率理论，利用历史数据来估计依赖于时间的过程到达边界所需的时间。他的初步工作在这个问题的研究和通过使用所涉及的过程的平均(或预期)行为来研究非随机函数跨越边界的情况之间提供了有用的联系。这项工作有可能对几个科学知识领域产生深远影响，在这些领域，越境时间很重要，有关过程行为的历史知识也是可用的。具体应用领域的例子包括等待时间的近似值：1)地震发生，2)下一次龙卷风到来，3)大坝溢出，4)计算机崩溃。所有这一切都是在有历史记录的情况下发生的。第二个问题涉及建立结果，将涉及自变量的近似的效率联系起来，以获得因变量的类似近似，这可能是在相互竞争的统计程序之间进行选择的一种节省成本的手段。第三个问题涉及基于因变量或过程的自归一化统计的研究。这种类型的统计提供了在制定参数的可信区间时使用的关键量。依赖如何进入这类问题的一个例子是，在选民不是独立行动而是相互影响决定的情况下，对投票给给定候选人的选民百分比进行区间估计。

The principal investigator (PI) considers problems which involve sums of dependent variables and whose solutions require the use of existing and new results of the theory of decoupling. Decoupling theory has been very successful in translating problems involving dependent variables into ones involving independent variables through the use of inequalities specially designed for that purpose. In this research the PI will study three major problems, and consider approaches (including decoupling methods) that may be used in solving them. The first problem involves a general and natural approach to approximate the expected waiting time for a random process to hit a boundary in terms of the function determined by the expected value of the supremum of the process. This approach permits the use of the historical averages in developing useful approximations. The second problem concerns an approximation to the speed of convergence of sums of dependent variables in terms of known results for sums of independent variables. This problem has the potential of widely expanding the applicability of the principle of conditioning, a method that extends the martingale central limit theorem. The third problem develops probability approximations for self-normalized martingales. Results from this problem have the potential to impact several areas of probability and statistics, including the theory of hypothesis testing and confidence intervals, since typically self-normalized variables serve as pivotal quantities in the construction of confidence intervals and tests of hypothesis. A common assumption made in the development of statistical and probabilistic methods concerns the predicate that the variables or processes are independent or have independence properties, that is, what happens to one of the variables does not affect what happens to the other. This independence assumption has the effect of simplifying the mathematical complexity of the problem and is quite common in practice. Due to this simplification it seems natural to seek approaches that would permit the use of results involving independent variables in the case the variables are dependent. The PI has been heavily involved in such a project through the development of ``Decoupling'' theory which has the above paradigm as its goal. His first problem in the current research is to develop a probabilistic theory for the use of historical data to obtain estimates of the time dependent processes take to reach a boundary. His preliminary work provides useful connections between the study of this problem and the study of the case of boundary crossing by non-random functions through the use of the average (or expected) behavior of the processes involved. This work has the potential to impact profoundly several areas of scientific knowledge where boundary crossing times are important and historical knowledge of the behavior of the process in question is available. Examples of specific areas of application include approximations to waiting time for: 1) an earthquake to happen, 2) for the next tornado to arrive, 3) for a dam to overflow, 4) for a computer to crash. All this when historical records are available. The second problem involves the development of results connecting the efficiency of approximations used involving independent variables to obtain similar approximations for dependent variables, a potentially cost saving device for choosing between competing statistical procedures. The third problem involves the study of self-normalized statistics based on dependent variables or processes. This type of statistic provides key quantities used in the development of confidence intervals for parameters. An example of how dependence enters into this type of problem concerns the development of interval estimates for the percentage of voters which will vote for a given candidate in the case the voters do not act independently but influence each other's decisions.