Mathematical Sciences: Large Devations and Related Topics

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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.