Regularization lemmas and convergence in total variation.

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 which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from a smooth (Wasserstein e.g.) distance to the total variation in a quantitative way. We provide a result removing the costly assumption that some non-degeneracy is required along the whole sequence, as we require only non-degeneracy at the limit. The price to pay is to control the smooth distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limits as their Malliavin matrix is deterministic. We show some applications of this result.
From joint papers with Vlad Bally and Guillaume Poly.