Mathematical Sciences: Probability Measures on Vector Spaces; Basic Results and Application

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This continues research involving limit theorems for real and vector valued random variables. Problems related to the central limit theorem and the law of the iterated logarithm motivated by trimming and self-normalizations will be studied. Limit sets for random samples of processes, as well as coverage problems will be examined. A primary focus will be to further examine the link between the small ball problem for Gaussian measures and metric entropy problems. Empirical processes and precise large deviation probabilities will also be considered. A central theme in probability, and throughout its diverse applications to problems in engineering, physics, and statistics, is the study of limit theorems for partial sums of independent random variables in both finite and infinite dimensional settings. This research place emphasis on the infinite dimensional situation and involves a blend of functional analysis and probability theory. Current work has fallen into several different areas, and includes small ball and large deviation probabilities for Gaussian measures, limit sets for random samples and related coverage problems, and Gaussian chaos applications to multiple Wiener-Ito stochastic integrals.