Dehling and Taqqu established the weak convergence of the empirical process for a long-range dependent stationary sequence under Gaussian subordination. We show that the corresponding density process, based on kernel estimators of the marginal density, converges weakly with the same normalization to the derivative of their limiting process. The phenomenon, which carries on for higher derivatives and for functional laws of the iterated logarithm, is in contrast with independent or weakly dependent situations, where the density process cannot be tight in the usual function spaces with supremum distances.
"Density Estimation Under Long-Range Dependence." Ann. Statist. 23 (3) 990 - 999, June, 1995. https://doi.org/10.1214/aos/1176324632