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May, 1985 Noncentral Limit Theorems for Quadratic Forms in Random Variables Having Long-Range Dependence
Robert Fox, Murad S. Taqqu
Ann. Probab. 13(2): 428-446 (May, 1985). DOI: 10.1214/aop/1176993001

Abstract

We study the weak convergence in D[0,1] of the quadratic form j=1[Nt]k=1[Nt]ajkHm(Xj)Hm(Xk), adequately normalized. Here as,<s< is a symmetric sequence satisfying |as|<,Hm is the mth Hermite polynomial and {Xj},j1, is a normalized Gaussian sequence with covariances rkkDL(k) as k, where 0<D<1 and L is slowly varying. We prove that, for all m1, the limit is Brownian motion when 1/2<D<1 and it is the non-Gaussian Rosenblatt process when 0<D<1/2.

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Robert Fox. Murad S. Taqqu. "Noncentral Limit Theorems for Quadratic Forms in Random Variables Having Long-Range Dependence." Ann. Probab. 13 (2) 428 - 446, May, 1985. https://doi.org/10.1214/aop/1176993001

Information

Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0569.60016
MathSciNet: MR781415
Digital Object Identifier: 10.1214/aop/1176993001

Subjects:
Primary: 60F05
Secondary: 33A65 , 60G10

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

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 2 • May, 1985
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