Mathematical Sciences: Limit Theorems for Function-Indexed Empirical Processes

这项研究将考虑的第一个问题是， 经验过程的极限定理 相当大的进展 近年来，为了解决 概率测度的充分必要条件 函数类，其相应的经验 过程满足中心极限定理。 其中一些 调查员自己也取得了进展。 他将继续 寻找问题的最终解决方案。 第二个问题是关于迭代法的 Banach空间值随机变量 一组三 条件被发现是必要和充分的，但一个 这些条件相当于证明一个较弱的版本， 原始问题。 调查人员将攻击这一减少 问题. 研究的第三部分和较小的部分将涉及 渗流理论 将审议的议题包括 渗流概率的连续性，临界概率的界 指数和期望簇大小的有限性 包含零。

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.