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

Abstract

Consider a Gaussian stationary sequence with unit variance $X=\{X_{k};k\in\mathbb{N}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_{n}=n^{-1/2}\sum^{n-1}_{k=0}f(X_{k})$, where $f$ designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of $V_{n}$ towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of $X$.