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

Abstract

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