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

主要研究者（PI）认为，涉及因变量的总和，其解决方案需要使用现有的和新的结果解耦理论的问题。解耦理论已经非常成功地通过使用专门设计的不等式将涉及因变量的问题转化为涉及自变量的问题。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）。第一个问题涉及到一个一般的和自然的方法来近似的期望等待时间的随机过程中的过程的上确界的期望值所确定的函数的边界。这种方法允许使用历史平均数来制定有用的近似值。第二个问题涉及的因变量和的收敛速度的近似已知的结果为自变量和。这个问题有可能广泛扩大适用性的原则条件，一种方法，扩展了鞅中心极限定理。第三个问题发展了自正规化鞅的概率近似。这个问题的结果有可能影响概率和统计学的几个领域，包括假设检验和置信区间理论，因为通常自归一化变量在置信区间和假设检验的构建中充当关键量。在统计和概率方法的发展过程中，一个常见的假设是变量或过程是独立的或具有独立性的，也就是说，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，并且在实践中非常常见。由于这种简化，似乎很自然地寻求方法，将允许使用的结果，涉及独立的变量的情况下，变量是相依的。PI通过发展以上述范式为目标的“解耦”理论，积极参与了这一项目。他在目前的研究中的第一个问题是开发一种概率理论，用于使用历史数据来获得对时间依赖过程到达边界的估计。他的初步工作提供了有用的联系之间的研究这个问题和研究的情况下，边界跨越非随机功能通过使用的平均（或预期）行为的过程中所涉及的。这项工作有可能深刻影响几个领域的科学知识，其中跨越边界的时间是重要的，历史知识的行为的过程中的问题是可用的。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）下一个龙卷风到来，3）大坝溢出，4）计算机崩溃。所有这些都是在有历史记录的情况下进行的。第二个问题涉及的发展结果连接的效率，使用的近似涉及自变量，以获得类似的近似因变量，一个潜在的节省成本的设备之间的竞争的统计程序进行选择。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。这种类型的统计量提供了用于确定参数置信区间的关键量。一个例子是如何进入这类问题的依赖关系的发展区间估计的百分比，将投票给一个给定的候选人的情况下，选民不独立行动，但影响对方的决定。

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.