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2020 Regularization lemmas and convergence in total variation
Vlad Bally, Lucia Caramellino, Guillaume Poly
Electron. J. Probab. 25: 1-20 (2020). DOI: 10.1214/20-EJP481


We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_{n})$ which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from smooth Wasserstein distances to total variation in a quantitative way. This is a well studied topic and one can consult for instance [3, 11, 14, 20] and the references therein for an overview of this issue. Each of the aforementioned references share the fact that some non-degeneracy is required along the whole sequence. We provide here the first result removing this costly assumption as we require only non-degeneracy at the limit. The price to pay is to control the smooth Wasserstein distance between the Malliavin matrix of the sequence and its limit, which is particularly easy in the context of Gaussian limit as the Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main findings of [19]. Another application concerns the approximation of the semi-group of a diffusion process by the Euler scheme in a quantitative way and under the Hörmander condition.


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Vlad Bally. Lucia Caramellino. Guillaume Poly. "Regularization lemmas and convergence in total variation." Electron. J. Probab. 25 1 - 20, 2020.


Received: 7 October 2019; Accepted: 8 June 2020; Published: 2020
First available in Project Euclid: 4 July 2020

zbMATH: 1444.60009
MathSciNet: MR4119120
Digital Object Identifier: 10.1214/20-EJP481

Primary: 60B110 , 60H07

Keywords: abstract Malliavin calculus , regularization techniques , total variation distance , Wasserstein distance


Vol.25 • 2020
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