DECOUPLING AND KHINTCHINES INEQUALITIES FOR U-STATISTICS

In this paper we introduce a fairly general decoupling inequality for U-statistics. Let {X(i)} be a sequence of independent random variables in a measurable space (S, F), and let (X(i)} be an independent copy of {X(i)}. Let PHI(x) be any convex increasing function for x greater-than-or-equal-to 0. Let PI(ij) be families of functions of two variables taking (S x S) into a Banach space (D, parallel-to . parallel-to). If the f(ij) is-an-element-of PI(ij) are Bochner integrable and [GRAPHICS] then, under measurability conditions, [GRAPHICS] where f = (f(ij), 1 less-than-or-equal-to i not-equal j less-than-or-equal-to n) and PI = (PI(ij), 1 less-than-or-equal-to i not-equal j less-than-or-equal-to n). In the case where PI is a family of functions of two variables satisfying f(ij) = f(ji) and f(ij)(X(i), X(j)) = f(ij)(X(j), X(i)), the reverse inequality holds (with a different constant). As a corollary, we extend Khintchine's inequality for quadratic forms to the case of degenerate U-statistics. A new maximal inequality for degenerate U-statistics is also obtained. The multivariate extension is provided.