Mathematical Sciences: Large Devations and Related Topics

主要研究概率论方面的三个课题：(a)马尔可夫链和过程的向量值泛函的大偏差；(b)非马尔可夫平稳过程的大偏差；(c)马尔可夫链的banach -空间值泛函的迭代对数定律。大偏差理论是概率论的一个分支，涉及随机变量的极限。众所周知，在某些理想情况下，随机变量是独立且同分布的，序列的平均值收敛于单个随机变量的平均值（就统计学而言，样本平均值收敛于总体平均值）。大偏差理论表明，随着样本量的增加，样本平均值与总体平均值偏差较小的概率呈指数小。首席研究员正在研究更加困难的依赖随机变量的情况，其依赖结构遵循马尔可夫过程。这一理论在统计力学和排队网络中有许多应用。

The principal investigator will conduct research on three topics concerning probability theory: (a) large deviations for vector-valued functionals of Markov chains and processes; (b) large deviation for non-Markov stationary processes: (c) law of the iterated logarithm for Banach-space valued functionals of Markov chains. Large deviation theory is a branch of probability theory involving limits of random variables. It is well known that in certain idealized cases, where the random variables are independent and identically distributed, that the average of the sequence converges to the average of the individual random variable (in terms of statistics, that the sample average converges to the population average). Large deviation theory shows that the probability that the sample average deviates from the population average by a small amount is exponentially small as the sample size increases. The principal investigator is working in the much more difficult case of dependent random variables, whose dependence structure follows a Markov process. There are many applications of this theory to statistical mechanics and queueing networks.