## Density estimation under long-range dependence.(English)Zbl 0843.62037

Summary: H. Dehling and M. S. Taqqu [Stat. Probab. Lett. 7, No. 1, 81-85 (1988; Zbl 0666.60031)] 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 the 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.

### MSC:

 62G07 Density estimation 62M99 Inference from stochastic processes 62G20 Asymptotic properties of nonparametric inference 60F17 Functional limit theorems; invariance principles

Zbl 0666.60031
Full Text: