BIVARIATE SYMMETRICAL STATISTICS OF LONG-RANGE DEPENDENT OBSERVATIONS

Let (X(j))j infinity = 1 be a stationary, mean-zero Gaussian sequence with covariances r(k) = EX(k+1)X1 satisfying r(0) = 1 and r(k) = k-D L(k) where D is small and L is slowly varying at infinity. Consider the sequence Y(j) = G(X(j)), j = 1,2,..., where G is any measurable function. We obtain the asymptotic distribution of certain degenerate von Mises and U-statistics based on the Y(j). As applications, we consider the sample variance, the chi-2-squared goodness of fit test and the Cramer-von Mises-Smirnov omega-2 criterion. The results are non-standard.