Noncentral Limit Theorems for Quadratic Forms in Random Variables Having Long-Range Dependence

Abstract

We study the weak convergence in $D[0,1]$ of the quadratic form $\sum _{j=1}^{[Nt]}\sum _{k=1}^{[Nt]}{a}_{j-k}{H}_m(X_j)H_m(X_k)$, adequately normalized. Here $a_s,-\mathrm{\infty }<s<\mathrm{\infty }$ is a symmetric sequence satisfying $\sum |a_s|<\mathrm{\infty },H_m$ is the $m$th Hermite polynomial and $\{X_j\},j\ge 1$, is a normalized Gaussian sequence with covariances $r_k\sim k^{-D}L(k)$ as $k\to \mathrm{\infty }$, where $0<D<1$ and $L$ is slowly varying. We prove that, for all $m\ge 1$, the limit is Brownian motion when $1/2<D<1$ and it is the non-Gaussian Rosenblatt process when $0<D<1/2$.