The Annals of Statistics

Density Estimation Under Long-Range Dependence

Sandor Csorgo and Jan Mielniczuk

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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.

Article information

Ann. Statist., Volume 23, Number 3 (1995), 990-999.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62M99: None of the above, but in this section 60F17: Functional limit theorems; invariance principles

Long-range dependence Gaussian subordination kernel density estimators weak convergence in supremum norm degenerate limiting processes Hermite polynomials


Csorgo, Sandor; Mielniczuk, Jan. Density Estimation Under Long-Range Dependence. Ann. Statist. 23 (1995), no. 3, 990--999. doi:10.1214/aos/1176324632.

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