On the distances between probability density functions

Vlad Bally (Université Paris-Est Marne-la-Vallée)
Lucia Caramellino (Università di Roma Tor Vergata)


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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Pages: 1-33

Publication Date: December 11, 2014

DOI: 10.1214/EJP.v19-3175


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