Simple bounds for convergence of empirical and occupation measures in 1- Wasserstein distance

Electronic Journal of Probability

Volumn 16, Issue , 2011, Pages 2296-2333

  Boissard, Emmanuel ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Transport inequalities; Uniform deviations

Indexed keywords

EID: 83255163943     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v16-958     Document Type: Article

References (35)

1. M. Ajtai, J. Komlos, and G. Tusnady. On optimal matchings. Combinatorica, 4(4):259-264, 1984.
2. P Baldi, G. Ben Arous, and G. Kerkyacharian. Large deviations and the Strassen theorem in Holder norm. Stochastic processes and their applications, 42(1):171-180, 1992.
3. F. Barthe and C. Bordenave. Combinatorial optimization over two random point sets. Arxiv preprint arXiv:1103.2734, 2011.
4. S. Bobkov and M. Ledoux. Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probability Theory and Related Fields, 107(3):383-400, 1997.
5. S.G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. Journal of Functional Analysis, 163(1):1-28, 1999.
6. E. Boissard and T.L. Gouic. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Arxiv preprint arXiv:1105.5263, 2011.
7. F. Bolley. Quantitative concentration inequalities on sample path space for mean field interaction. ESAIM: PS, July 2010, Vol. 14, p. 192-209, 2010.
8. F. Bolley, A. Guillin, and C. Villani. Quantitative concentration inequalities for empirical measures on non-compact spaces. Probability Theory and Related Fields, 137:541-593, 2007.
9. F. Bolley and C. Villani. Weighted Csiszar-Kullback-Pinsker inequalities and applications to transportation inequalities. In Annales Faculté des Sciences Toulouse Mathématiques, volume 14, page 331. Université Paul Sabatier, 2005.
10. E. Del Barrio, E. Giné, and C. Matran. Central limit theorems for the Wasserstein distance between the empirical and the true distributions. Annals of Probability, 27(2):1009-1071, 1999.
11. A. Dembo and O. Zeitouni. Large Deviations Techniques Amd Applications. Jones and Bartlett, 1993.
12. S. Dereich, F. Fehringer, A. Matoussi, and M. Scheutzow. On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces. Journal of Theoretical Probability, 16(1):249-265, 2003.
13. V. Dobric and J.E. Yukich. Exact asymptotics for transportation cost in high dimensions. J. Theoretical Prob, pages 97-118, 1995.
14. R.M. Dudley. The speed of mean Glivenko-Cantelli convergence. The Annals of Mathematical Statistics, 40(1):40-50, 1969.
15. F. Fehringer. Kodierung von Gaußmaßen. PhD thesis, 2001.
16. N. Gozlan and C. Léonard. A large deviation approach to some transportation cost inequalities. Probability Theory and Related Fields, 139:235-283, 2007.
17. N. Gozlan and C. Léonard. Transport inequalities. A survey. Markov Processes and Related Fields 16 (2010) 635-736, 2010.
18. S. Graf, H. Luschgy, and G. Pages. Functional quantization and small ball probabilities for Gaussian processes. Journal of Theoretical Probability, 16(4):1047-1062, 2003.
19. A. Guillin H. Djellout and L. Wu. Transportation cost-information inequalities for random dynamical systems and diffusions. Annals of Probability, 32:2702-2732, 2004.
20. A. Joulin and Y. Ollivier. Curvature, concentration and error estimates for Markov chain Monte Carlo. The Annals of Probability, 38(6):2418-2442, 2010.
21. J. Kuelbs and W.V Li. Metric entropy and the small ball problem for Gaussian measures. Journal of Functional Analysis, 116(1):133-157, 1993.
22. M. Ledoux. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics, pages 165-294, 1996.
23. M. Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 208:2005-2006, 2001.
24. W.V. Li and W Linde. Approximation, metric entropy and small ball estimates for Gaussian measures. The Annals of Probability, 27(3):1556-1578, 1999.
25. K. Marton. Bounding d-distance by informational divergence: a method to prove measure concentration. The Annals of Probability, 24(2):857-866, 1996.
26. P Massart. The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. The Annals of Probability, 18(3):1269-1283, 1990.
27. P Massart. Concentration inequalities and model selection, volume 1896 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003.
28. Y. Ollivier. Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis, 256(3):810-864, 2009.
29. M. M. Rao and Z. D. Ren. Theory ofOrlicz spaces, volume 146 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1991.
30. D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.
31. M. Talagrand. Matching random samples in many dimensions. The Annals of Applied Probability, 2(4):846-856, 1992.
32. A.W. Van der Vaart and J.A. Wellner. Weak convergence and empirical processes. Springer Verlag, 1996.
33. VS. Varadarajan. On the convergence of sample probability distributions. Sankhya: The Indian Journal of Statistics, 19(1):23-26, 1958.
34. C. Villani. Optimal transport, old and new, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009.
35. R. Wang, X. Wang, and L. Wu. Sanov’s theorem in the Wasserstein distance: A necessary and sufficient condition. Statistics & Probability Letters, 80(5-6):505-512, 2010.