The Annals of Probability

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

Robert Fox and Murad S. Taqqu

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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 $0 < D < 1/2$.

Article information

Source
Ann. Probab., Volume 13, Number 2 (1985), 428-446.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993001

Digital Object Identifier
doi:10.1214/aop/1176993001

Mathematical Reviews number (MathSciNet)
MR781415

Zentralblatt MATH identifier
0569.60016

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 33A65

Keywords
Weak convergence Brownian motion Rosenblatt process Hermite polynomials Wiener multiple integrals long-range dependence fractional Gaussian noise fractional ARMA

Citation

Fox, Robert; Taqqu, Murad S. Noncentral Limit Theorems for Quadratic Forms in Random Variables Having Long-Range Dependence. Ann. Probab. 13 (1985), no. 2, 428--446. doi:10.1214/aop/1176993001. https://projecteuclid.org/euclid.aop/1176993001


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