Mathematical Sciences: Topics in Large Deviations

9627045德·阿科斯塔研究员建议对与一些重要的随机序列或假设有关的大偏差基本问题的研究作出贡献。这包括开发适当的抽象数学工具，识别自然地导致有趣的大偏差问题的模型，以及计算特定速率函数的解析表达式。该项目的范围包括以下情况：(1)马尔可夫过程的轨迹；(2)某些类半鞅的轨迹；(3)各种依赖条件下的经验度量。概率论及其应用的一个基本主题是研究由随机序列构成的平均序列在某种意义上收敛于确定性极限的情况。大偏差理论涉及估计平均值偏离极限所代表的“典型行为”的小概率的问题。这里有两种情况与拟议的研究具有直接或潜在的相关性：(A)概率模型越来越多地被用于分析复杂通信系统和计算机网络的性能。大偏差理论(特别是某些马尔可夫过程的大偏差)是分析这些类型的大系统中罕见事件(即与“正常行为”的偏差)的有力工具。(B)在统计力学的几个重要模型中，大偏差理论在分析与热力学极限有关的罕见事件时非常有用。

9627045 de Acosta The investigator proposes to contribute to the study of basic questions in Large Deviations related to some important stochastic sequences or ppocesses. This includes the development of suitable abstract mathematical tools, the identification of models which lead naturally to interesting large deviation questions, and the computation of the analytical expression of specific rate functions. The scope of the project includes the following situations: (i) trajectories of Markov processes (ii) trajectories of certain classes of semimartingales (iii) empirical measures under various dependence conditions. A fundamental theme in Probability Theory and its applications is the study of situations in which a sequence of averages constructed from a stochastic sequence converges in some sense to a deterministic limit. Large Deviation theory deals with the problem of estimating the small probabilities that the averages will deviate from the "typical behavior" represented by the limit. Here are two situations to which the proposed research has direct or potential relevance; (a) Probabilistic models are increasingly used in analyzing the performance of complex communication systems and computer networks. Large Deviation theory (in particular, large deviations of certain Markov processes) is a powerful tool in the analysis of rare events (that is, deviations from "normal behavior") in large systems of those types. (b) Large Deviation theory is very useful in the analysis of rare events associated to thermodynamic limits in several important models in Statistical Mechanics.