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

Consider a Gaussian stationary sequence with unit variance X = {Xk;k E NU{0}}. Assume that the central limit theorem holds for a weighted sum of the form Vn = n.1/2-n-1 k=0 f (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.