Mathematical Sciences: Limit Theorems for Function-Indexed Empirical Processes

本研究将考虑的第一个问题是经验过程的极限定理。近年来，在解决概率测度和相应经验过程满足中心极限定理的函数类的充要条件问题方面取得了相当大的进展。其中一些进步是调查者自己取得的。他将继续寻找问题的最终解决办法。第二个问题涉及到巴拿赫空间值随机变量的迭代对数律。一组三个条件被发现既必要又充分，但其中一个条件相当于证明了原始问题的弱版本。研究者将解决这个简化的问题。研究的第三部分和较小的部分将涉及渗透理论。其中要考虑的主题将是渗透概率的连续性，临界指数的界限和包含零的簇的期望大小的有限性。

The first problem this research will consider is that of limit theorems for empirical processes. Considerable progress has been made in recent years towards solving the problem of necessary and sufficient conditions on probability measures and classes of functions for which the corresponding empirical process satisfies the central limit theorem. Some of these advances were made by the investigator himself. He will continue to look for the final solution of the problem. The second problem concerns the law of iterated logarithms for Banach space valued random variables. A set of three conditions is found to be both necessary and sufficient but one of these conditions amounts to proving a weaker version of the original problem. The investigator will attack this reduced problem. The third and smaller part of the research will deal with percolation theory. Among the topics to be considered will be continuity of the percolation probability, bounds on critical exponents and finiteness of the expected size of cluster containing zero.