C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Article published in Annals of Statistics, vol. 39, n. 3, p. 1399-1426, 2011. Download a pdf version.

Let be a stationary mean-zero Gaussian process with covariances satisfying:

and where D is in (0,1) and L is slowly varying at infinity. Consider the U-process defined as

where I is an interval included in and G is a symmetric function. In this paper, we provide central and non-central limit theorems for . They are used to derive, in the long-range dependence setting, new properties of many well-known estimators such as the Hodges- Lehmann estimator, which is a well-known robust location estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated robust scale estimator. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators. The limiting distributions are expressed through multiple Wiener-Itô integrals.