60

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# Bivariate symmetric statistics of
long-range dependent observations

**Herold Dehling
**

Murad S. Taqqu

### Abstract:

*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.