Distributional limit theorems over a stationary Gaussian sequence of random vectors

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