Mathematical Sciences: Applications and Refinements of the Theory of Large Deviations

这项研究集中于大偏差理论的扩展和改进，特别是在通信和控制问题、分子(DNA和蛋白质)序列的比较分析、排队过程和序贯实验的研究中的应用。在预期的结果中，有新的分布极限定理，以及通过适当的停止时间选择的马尔可夫过程和马尔可夫随机场的经验法则的新分析。另一种有希望的新方法是基于使用(FKG类型的)结合不等式和交换性作为获得大偏差界的工具。这些界适用于时变参数的估计和稀有事件概率抽样方案的比较。这项研究的中心是估计罕见事件发生的可能性，这些事件发生在具有根本重要性的地区。例如，评估异常DNA序列片段组成的统计意义，分析复杂排队系统中的长等待时间，以及分析并行计算机系统。

This research centers on extensions and refinements of large deviations theory, especially motivated by applications to problems of communication and control, to comparative analysis of molecular (DNA and protein) sequences, and to studies of queueing processes and sequential experimentation. Among the expected results are new limit theorems for distributions conditioned on rare events of exponentially small probabilities, and the novel analysis of the empirical law of segments of Markov processes and Markov random fields that are selected by means of appropriate stopping times. Another promising new approach is based on using association inequalities (of the FKG type), and exchangeability as tools for obtaining large deviations bounds. These bounds are applicable for the estimation of time-varying parameters and for comparing sampling schemes in terms of rare events probabilities. This research centers on estimating the chances for rare events to occur, in areas where such events are of fundamental importance. Examples include assessing statistical significance of unusual DNA sequence segment compositions, analysis of long waiting times in complex queueing systems, and analysis of parallel computer systems.