ORDER OF MAGNITUDE BOUNDS FOR EXPECTATIONS OF DELTA(2)-FUNCTIONS OF NONNEGATIVE RANDOM BILINEAR-FORMS AND GENERALIZED U-STATISTICS

Let X-1, Y-1, Y-2,..., X-n, Y-n be independent nonnegative rv's and le t {b(ij)}(1 less than or equal to i, j less than or equal to n) be an array of nonnegative constants. We present a method of obtaining the o rder of magnitude of E Phi (Sigma(1 less than or equal to i, j less th an or equal to n) b(ij) X-i Y-j), for any such {X-i}, {Y-i} and {b(ij) } and any nondecreasing function Phi on [0, infinity) with Phi(0) = 0 and satisfying a Delta(2) growth condition Furthermore, this technique is extended to provide the order of magnitude of E Phi (Sigma(1 less than or equal to i, j less than or equal to n) b(ij) X-i Y-j), where { f(ij)(x, y)}(1 less than or equal to i, j less than or equal to n) is any array of nonnegative functions. Far arbitrary functions {g(ij)(x,y )}(1 less than or equal to i not equal j less than or equal to n), the aforementioned approximation enables us to identify the order of magn itude of E Phi (/Sigma (1 less than or equal to i not equal j less tha n or equal to n) g(ij)(X-i, X-j)/) whenever decoupling results and Khi ntchine-type inequalities apply, such as Phi is convex, L(g(ij)(X-i, X -j)) = L(g(ji)(X-j, X-i)) and Eg(ij)(X-i, x) = 0 for all x in the rang e of X-j.