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

Robert Fox and Murad S. Taqqu
Source: Ann. Probab. Volume 13, Number 2 (1985), 428-446.

#### 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 $0 < D < 1/2$.

First Page:
Primary Subjects: 60F05
Secondary Subjects: 60G10, 33A65
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1176993001
Digital Object Identifier: doi:10.1214/aop/1176993001
Mathematical Reviews number (MathSciNet): MR781415
Zentralblatt MATH identifier: 0569.60016