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 $V_n=n^{-1/2}{\sum }_{k=0}^{n-1}f(X_k)$, 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 $X$.

Published at http://dx.doi.org/10.3150/14-BEJ646 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Central limit theorems for sequences of multiple stochastic integrals
• Convergence of densities of some functionals of Gaussian processes

Cited by in corpus (2)

• Superconvergence phenomenon in Wiener chaoses
• Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables