Noncentral limit theorems for quadratic forms in random variables having long-range dependence. (English) Zbl 0569.60016

Fox, Robert; Taqqu, Murad S.

Ann. Probab. 13, 428-446 (1985).

Let \(X_ j\), \(j\geq 1\), be a stationary Gaussian sequence satisfying \(EX_ j=0\) and \(EX^ 2_ j=1\), and suppose that there is a constant \(0<D<1\) and a slowly varying function L such that the covariances \(r_ k=EX_ jX_{j+k}\sim k^{-D}L(k)\) as \(k\to \infty\). Let m be a positive integer and \(H_ m\) be the mth Hermite polynomial with leading coefficient 1. The authors prove that for \(0<D<\) and all \(m\geq 1\) the adequately normalized quadratic form \[ Z_ N(t)=(1/N^{1- D}L(N))\{\sum^{[Nt]}_{j=1}\sum^{[Nt]}_{k=1}a_{j-k}H_ m(X_ j)H_ m(X_ k)+ \]
\[ - E\sum^{[Nt]}_{j=1}\sum^{[Nt]}_{k=1}a_{j-k}H_ m(X_ j)H_ m(X_ k)\} \] converges weakly in D[0,1] to the non-Gaussian Rosenblatt process, whereas for \(<D<1\) and all \(m\geq 1\) the quadratic form \(\bar Z_ N(t)=[L(N)/N^{D-}]Z_ N(t)\), similarly normalized, converges weakly in D[0,1] to the Brownian motion. The proofs make use of the spectral representation of \(X_ j\) and the Wiener Itô-Dobrushin representation of \(H_ m(X_ j)\), respectively.

Reviewer: L.Hahn

MSC:

60F05	Central limit and other weak theorems
60G10	Stationary stochastic processes
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

limit theorems for quadratic forms; stationary Gaussian sequence; Hermite polynomial; Wiener Itô-Dobrushin representation