THE LAW OF THE ITERATED LOGARITHM AND MARCINKIEWICZ LAW OF LARGE NUMBERS FOR B-VALUED U-STATISTICS

Suppose that B is a separable Banach space and (S, J, P) a probability space. H is a measurable symmetric kernel function from S-m into B. I n this paper we shall further study some limit theorems for B-valued U -statistics (UmH)-H-n based on P and H. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterat ed logarithm. Our results can be regarded as extensions of correspondi ng results for sums of independent B-valued random variables to U-stat istics.