Weak Convergence of the Sequential Empirical Process of some Long-Range Dependent Sequences with Respect to a Weighted Norm

Abstract

Let $(X_k{)}_{k\ge 1}$ be a Gaussian long-range dependent process with $E{X}_1=0$, $E{X}_1^2=1$ and covariance function $r(k)=k^{-D}L(k)$. For any measurable function $G$ let $(Y_k{)}_{k\ge 1}=(G(X_k){)}_{k\ge 1}$. We study the asymptotic behaviour of the associated sequential empirical process $\left(R_N(x,t)\right)$ with respect to a weighted norm $\Vert \cdot {\Vert }_w$. We show that, after an appropriate normalization, $\left(R_N(x,t)\right)$ converges weakly in the space of càdlàg functions with finite weighted norm to a Hermite process.

References in corpus (1)

• Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes