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.
