Electronic Journal of Probability

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

Emmanuel Boissard

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Uniform deviations Transport inequalities

This work is licensed under aCreative Commons Attribution 3.0 License.


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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