ASYMPTOTIC PROPERTIES OF U-PROCESSES UNDER LONG-RANGE DEPENDENCE

Let (Xi)(i >= 1) be a stationary mean-zero Gaussian process with covariances rho(k) = E(X-1 Xk+1) satisfying rho(0) = 1 and rho(k) = k(-D) L(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {U-n(r), r is an element of 1} defined as U-n(r) = 1/n (n-1) Sigma(1 <= i not equal j <= n) 1{G(X-i, X-j)<= r} where I is an interval included in R, and G is a symmetric function. In this paper, we provide central and noncentral limit theorems for U-n. 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-Ito integrals.