Electronic Journal of Probability

Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance

Emmanuel Boissard

Full-text: Open access

Abstract

We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the reference measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a contracting Markov chain in 1-Wasserstein distance are also given. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 83, 2296-2333.

Dates
Accepted: 15 November 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820252

Digital Object Identifier
doi:10.1214/EJP.v16-958

Mathematical Reviews number (MathSciNet)
MR2861675

Zentralblatt MATH identifier
1254.60014

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 39B72: Systems of functional equations and inequalities

Keywords
Uniform deviations Transport inequalities

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Boissard, Emmanuel. Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance. Electron. J. Probab. 16 (2011), paper no. 83, 2296--2333. doi:10.1214/EJP.v16-958. https://projecteuclid.org/euclid.ejp/1464820252


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