Probability Measures on Vector Spaces: Theory and Applications

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Applications of probability in modern science frequently involve the study of random quantities with many components (dimensions), or perhaps even of a geometric nature. Thus they require probability estimates and limit theorems which are applicable to random sets, or which are dimension free (hence, in essence, infinite dimensional). A major theme in the investigator's previous work, and in much of the current research, addresses both of these issues in a variety of settings. As a first example consider the link between small ball probabilities and metric entropy problems, which showed certain probability estimates are equivalent to problems in approximation theory. This link led to the solution of a long standing problem in approximation theory, and portions of the proposed research involve important unsolved analogues of this problem. Another example is the study of the Gibbs conditioning principle of statistical mechanics for statistics with infinitely many components. To begin to handle this type of problem one needs non-logarithmic estimates of large deviation probabilities which are dimension free. These estimates depend critically on dominating points and a suitable representation formula for the probabilities. A variety of conditional limit theorems are to be considered. Additional problems exhibiting these general features are also proposed, and connect with classical geometry, analysis, and statistics. This work includes further non-logarithmic large deviation probabilities for partial sums of independent random vectors, an investigation of dominating points in a more general setting, and the application of these results to conditional limit theorems for infinite dimensional statistics. Limit sets for random samples of stochastic processes, as well as related coverage problems will be examined, and a primary focus will be to further examine the link between small ball probabilities and non-classical functional laws of the iterated logarithm applicable to occupation measures. Problems concerning vector valued partial sums, cluster sets, small ball probabilities, self-normalized partial sums, and limit theorems for convex hulls of Brownian motion paths are also to be considered.