A total variation version of Breuer--Major Central Limit Theorem under $\mathbb{D}^{1,2}$ assumption

TL;DR

This paper proves a total variation CLT for Gaussian sequences with functions in the Malliavin space ^{1,2}, extending previous results by weakening integrability conditions and removing Hermite rank restrictions.

Contribution

It extends the Breuer--Major CLT to a total variation setting under weaker Malliavin space assumptions and without restrictions on Hermite rank.

Findings

Establishes total variation convergence for a broad class of Gaussian functionals.

Reduces Malliavin integrability requirements from ^{1,4} to ^{1,2}.

Introduces a novel use of the sharp operator in the proof.

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.