DECOUPLING INEQUALITIES FOR THE TAIL PROBABILITIES OF MULTIVARIATE U-STATISTICS

In this paper we present a decoupling inequality that shows that multivariate U-statistics can be studied as sums of(conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get the following result: Let {X(j)} be a sequence of independent random variables on a measurable space (L, S) and let {X(i)((j))}, j = 1,..., k, be k independent copies of (X(i)). Let fi(1)i(2)...i(k) be families of functionsof k variables taking (S x ... x S) into a Banach space (B, I II). Then, for all n greater than or equal to k greater than or equal to 2, t > 0, there exist numerical constants C-k depending on k only so that GRAPHICS GRAPHICS The reverse bound holds if, in addition, the following symmetry condition holds almost surely: GRAPHICS for all permutations pi of (1,...,k).