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In this note, we establish a qualitative total variation version ofBreuer--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 acentered stationary Gaussian process, under the hypothesis that the function$f$ has Hermite rank $d \geq 1$ and belongs to the Malliavin space $\mathbbD^{1,2}$. This result in particular extends the recent works of [NNP21], wherea quantitative version of this result was obtained under the assumption thatthe function $f$ has Hermite rank $d= 2$ and belongs to the Malliavin space$\mathbb D^{1,4}$. We thus weaken the $\mathbb D^{1,4}$ integrabilityassumption to $\mathbb D^{1,2}$ and remove the restriction on the Hermite rankof the base function. While our method is still based on Malliavin calculus, weexploit a particular instance of Malliavin gradient called the sharp operator,which reduces the desired convergence in total variation to the convergence indistribution of a bidimensional Breuer--Major type sequence.

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