Bivariate symmetric statistics of long-range dependent observations

Herold Dehling
Murad S. Taqqu


Let be a stationary, mean-zero Gaussian sequence with covariances satisfying and where D is small and L is slowly varying at infinity. Consider the sequence , where G is any measurable function. We obtain the asymptotic distribution of certain degenerate von Mises and U-statistics based on the . As applications, we consider the sample variance, the -squared goodness of fit test and the Cramer-von Mises-Smirnov criterion. The results are non-standard.