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Article Dans Une Revue Electronic Journal of Probability Année : 2014

On the distance between probability density functions

Résumé

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.

Dates et versions

hal-00926401 , version 1 (09-01-2014)

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Vlad Bally, Lucia Caramellino. On the distance between probability density functions. Electronic Journal of Probability, 2014, 19 (110), pp.1-33. ⟨hal-00926401⟩
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