A total variation version of Breuer–Major Central Limit Theorem under D1,2 assumption

Abstract

In this note, we establish a qualitative total variation version of Breuer–Major Central Limit Theorem for a sequence of the type $\frac{1}{\sqrt{n}}{\sum }_{1\le k\le n}f(X_k)$, where ${(X_k)}_{k\ge 1}$ is a centered stationary Gaussian process, under the hypothesis that the function f has Hermite rank $d\ge 1$ 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.