Mathematical Sciences: Probabilistic Limit Theorems and Applications

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The investigator will conduct research on empirical processes, U-processes, their bootstrap and some of their applications in statistics. The limit theory for degenerate U- processes has not yet reached a satisfactory stage, and the bootstrap of U-processes, degenerate or not, is far from being resolved. The applications include asymptotic theory as well as uniqueness and robustness properties of the empirical simplicial median. Open questions for U-statistics, such as almost sure convergence to zero of sums of normalized products of random variables will also be considered. Modern empirical process theory, developed during the last twenty years and not yet totally completed, not only sheds light on one of the basic questions in Statistics (namely, how well do sample properties approach population properties) but has been providing a high road to the study of important problems in asymptotic statistics. U--processes, a non-linear generalization of empirical processes, should play a similar, although more restricted, role. We propose to continue work on empirical and U--processes and their bootstraps, and to investigate some applications, such as to multidimensional robust location parameters.