A limit result for U-statistics of binary variables

Define r/(k, n) = U-k,U- n - n(k/2)H(k)(Sigma(j =1)(n) X-j/root n), where U-k,U- n = Sigma(1) less than or equal to i(1) not equal ... not equal i(k) less than or equal to n X-i1...X-ik is a symmetric U-type statistic, H-k(.) is the Hermite polynomial of degree k, and {X, X-n, n greater than or equal to 1} are independent identically distributed binary random variables with Pr(X is an element of {-1, 1}) = 1. We show that [GRAPHICS] according as EX = 0 or EX not equal 0, respectively.