## Bernoulli

• Bernoulli
• Volume 21, Number 4 (2015), 2336-2350.

### 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 $X$.

#### Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2336-2350.

Dates
Revised: May 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1438777596

Digital Object Identifier
doi:10.3150/14-BEJ646

Mathematical Reviews number (MathSciNet)
MR3378469

Zentralblatt MATH identifier
1344.60025

#### Citation

Hu, Yaozhong; Nualart, David; Tindel, Samy; Xu, Fangjun. Density convergence in the Breuer–Major theorem for Gaussian stationary sequences. Bernoulli 21 (2015), no. 4, 2336--2350. doi:10.3150/14-BEJ646. https://projecteuclid.org/euclid.bj/1438777596

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