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Pré-Publication, Document De Travail Année : 2023

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

Federico Dalmao
  • Fonction : Auteur
Guillaume Poly
  • Fonction : Auteur

Résumé

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.
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Dates et versions

hal-04349914 , version 1 (18-12-2023)

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Jürgen Angst, Federico Dalmao, Guillaume Poly. A total variation version of Breuer--Major Central Limit Theorem under $\mathbb{D}^{1,2}$ assumption. 2023. ⟨hal-04349914⟩
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