Weak convergence of the weighted sequential empirical process of some long-range dependent data

Abstract

Let (X_k)k>=1 be a Gaussian long-range dependent process with EX_1 = 0, EX^2_1 1 = 1 and covariance function r(k) = k^(-D)L(k). For any measurable function G let (Y_k)k>= 1 = (G(X_k))k>= 1. We study the asymptotic behaviour of the associated sequential empirical process (R_N(x,t)) with respect to a weighted sup-norm ||*||w. We show that, after an appropriate normalization, (R_N(x,t)) converges weakly in the space of c adl ag functions with nite weighted norm to a Hermite process.