Mathematical Sciences: Probabilistic Limit Theorems and Applications

研究人员将对经验过程、U-过程、它们的自举及其在统计学中的一些应用进行研究。退化U-过程的极限理论还没有达到令人满意的阶段，U-过程的自举问题，无论是退化的还是非退化的，都远未解决。它的应用包括渐近理论以及经验单纯中值的唯一性和稳健性。U-统计量的公开问题，例如随机变量的归一化乘积和几乎必然收敛到零，也将被考虑。现代经验过程理论是近二十年来发展起来的，但尚未完全完成，它不仅揭示了统计学中的一个基本问题(即样本性质如何接近总体性质)，而且为研究渐近统计学中的重要问题提供了一条大道。U-过程是经验过程的非线性概括，应该发挥类似的作用，尽管受到更多的限制。我们建议继续研究经验过程和U-过程及其自举过程，并研究一些应用，如多维稳健位置参数。

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.