Distributional limit theorems over a stationary Gaussian sequence of random vectors

Abstract

Let { X j } j=1 ∞ be a stationary Gaussian sequence of random vectors with mean zero. We study the convergence in distribution of a n −1∑ j=1 n (G(X j)−E[G(X j)]), where G is a real function in R d with finite second moment and { a n } is a sequence of real numbers converging to infinity. We give necessary and sufficient conditions for a n −1∑ j=1 n (G(X j)−E[G(X j)]) 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 a n −1∑ j=1 n (G(X j)−E[G(X j)]) is determined by the first two terms in the Fourier expansion of G( x).