Density convergence in the Breuer-Major theorem for Gaussian stationary sequences

Abstract

Consider a Gaussian stationary sequence with unit variance X={Xk;k∈N∪{0}}. Assume that the central limit theorem holds for a weighted sum of the form Vn=n−1/2∑n−1k=0f(Xk), where f designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of Vn towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of X.