Electronic Journal of Probability

On the distances between probability density functions

Vlad Bally and Lucia Caramellino

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 110, 33 pp.

Accepted: 11 December 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

This work is licensed under a Creative Commons Attribution 3.0 License.


Bally, Vlad; Caramellino, Lucia. On the distances between probability density functions. Electron. J. Probab. 19 (2014), paper no. 110, 33 pp. doi:10.1214/EJP.v19-3175. https://projecteuclid.org/euclid.ejp/1465065752

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