## Abstract

In this note, we establish a qualitative total variation version of Breuer--Major Central Limit Theorem for a sequence of the type $\frac{1}{\sqrt{n}} \sum_{1\leq k \leq n} f(X_k)$, where $(X_k)_{k\ge 1}$ is a centered stationary Gaussian process, under the hypothesis that the function $f$ has Hermite rank $d \geq 1$ and belongs to the Malliavin space $\mathbb D^{1,2}$. This result in particular extends the recent works of [NNP21], where a quantitative version of this result was obtained under the assumption that the function $f$ has Hermite rank $d= 2$ and belongs to the Malliavin space $\mathbb D^{1,4}$. We thus weaken the $\mathbb D^{1,4}$ integrability assumption to $\mathbb D^{1,2}$ and remove the restriction on the Hermite rank of the base function. While our method is still based on Malliavin calculus, we exploit a particular instance of Malliavin gradient called the sharp operator, which reduces the desired convergence in total variation to the convergence in distribution of a bidimensional Breuer--Major type sequence.

## Full Text

### Topics from this Paper

- Hermite Rank
- Convergence In Total Variation
- Malliavin Calculus
- Convergence In Variation
- Qualitative Version + Show 5 more

Create a personalized feed of these topics

Get Started### Similar Papers

- Bernoulli
- May 1, 2018

- Electronic Journal of Probability
- Jan 1, 2013

- IEEE Transactions on Information Theory
- Jan 1, 1992

- Advances in Applied Probability
- Jun 1, 1994

- Advances in Applied Probability
- Jun 1, 1994

- Journal of Mathematical Sciences
- Aug 17, 2019

- Proceedings of the American Mathematical Society
- Mar 18, 2015

- Stochastic Processes and their Applications
- Mar 1, 1995

- Journal of Functional Analysis
- Mar 1, 2014

- SIAM Journal on Control and Optimization
- Mar 9, 2023

- Dec 14, 2021

- Statistics & Probability Letters
- May 1, 2019

- The Annals of Applied Probability
- Nov 1, 1997

- Journal of Applied Probability
- Jun 1, 1986