Distributional limit theorems over a stationary Gaussian sequence of random vectors

Miguel A. Arcones

Stochastic Processes and their Applications, 2000, vol. 88, issue 1, 135-159

Abstract: Let {Xj}j=1[infinity] be a stationary Gaussian sequence of random vectors with mean zero. We study the convergence in distribution of an-1[summation operator]j=1n (G(Xj)-E[G(Xj)]), where G is a real function in with finite second moment and {an} is a sequence of real numbers converging to infinity. We give necessary and sufficient conditions for an-1[summation operator]j=1n (G(Xj)-E[G(Xj)]) to converge in distribution for all functions G with finite second moment. These conditions allow to obtain distributional limit theorems for general sequences of covariances. These covariances do not have to decay as a regularly varying sequence nor being eventually nonnegative. We present examples when the convergence in distribution of an-1[summation operator]j=1n (G(Xj)-E[G(Xj)]) is determined by the first two terms in the Fourier expansion of G(x).

Keywords: Long-range; dependence; Stationary; Gaussian; sequence; Hermite; polynomials (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (1)

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