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

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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.