Probability Measures of Vector Spaces; Basic Results and Applications

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9703740 Kuelbs The principal investigator will continue his research in probability theory. It includes non-logarithmic large deviation probabilities for partial sums of independent random vectors via dominating points in the infinite dimensional setting, and also the application of these results to a Gibbs conditioning principle for certain infinite dimensional statistics. Limit sets for random samples of 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 metric entropy problems in approximation theory. For Gaussian measures, this linkage has been useful in a number of problem areas, but a more detailed examination from several points of view is proposed. Problems concerning vector valued partial sums, empirical processes, self-normalized partial sums, and limit theorems for convex hulls for Brownian motion are also to be considered. 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 currently proposed 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 estimat es of large deviation probabilities which are dimension free. Such estimates were obtained recently, and their application to Gibbs conditioning is being initiated. Additional problems exhibiting these general features are also to be considered, and connect with classical geometry, analysis, and statistics.