On the distance between probability density functions

Vlad Bally; Lucia Caramellino

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arXiv · 1311.7555

On the distance between probability density functions

Vlad Bally·Lucia Caramellino

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Abstract

We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in $L^{1}$ of the densities.

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Vlad Bally, Lucia Caramellino. 2013-11-29. On the distance between probability density functions. https://arxiv.org/abs/1311.7555

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