Open Access
November 2015 Density convergence in the Breuer–Major theorem for Gaussian stationary sequences
Yaozhong Hu, David Nualart, Samy Tindel, Fangjun Xu
Bernoulli 21(4): 2336-2350 (November 2015). DOI: 10.3150/14-BEJ646


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 $X$.


Download Citation

Yaozhong Hu. David Nualart. Samy Tindel. Fangjun Xu. "Density convergence in the Breuer–Major theorem for Gaussian stationary sequences." Bernoulli 21 (4) 2336 - 2350, November 2015.


Received: 1 March 2014; Revised: 1 May 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1344.60025
MathSciNet: MR3378469
Digital Object Identifier: 10.3150/14-BEJ646

Keywords: Breuer–Major theorem , density convergence , Gaussian stationary sequences , Malliavin calculus , moving average representation

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
Back to Top