## Electronic Journal of Probability

### Behavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditions

#### Abstract

We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 6, 32 pp.

Dates
Accepted: 14 January 2019
First available in Project Euclid: 14 February 2019

https://projecteuclid.org/euclid.ejp/1550113245

Digital Object Identifier
doi:10.1214/19-EJP266

Mathematical Reviews number (MathSciNet)
MR3916326

Zentralblatt MATH identifier
1406.60009

#### Citation

Dedecker, Jérôme; Merlevède, Florence. Behavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditions. Electron. J. Probab. 24 (2019), paper no. 6, 32 pp. doi:10.1214/19-EJP266. https://projecteuclid.org/euclid.ejp/1550113245

#### References

• [1] Ajtai, M., Komlos, J. and Tusnady, G.: On optimal matching. Combinatorica 4, (1983), 259-264.
• [2] Bach, F. and Weed, J.: Sharp asymptotic and finite sample rates of convergence of empirical measures in Wasserstein distance. (2017) arXiv:1707.00087v1
• [3] von Bahr, B. and Esseen, C.-G.: Inequalities for the $r$th absolute moment of a sum of random variables, $1 \leq r \leq 2$. Ann. Math. Statist. 36, (1965), 299-303.
• [4] del Barrio, E., Giné, E. and Matrán, C.: Central limit theorems for the Wasserstein distance between the empirical and the true distributions. Ann. Probab. 27, (1999), 1009-1071.
• [5] del Barrio, E., Giné, E. and Utzet, F.: Asymptotics for ${\mathbb L}^2$ functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances. Bernoulli 11, (2005), 131-189.
• [6] Barthe, F. and Bordenave, C.: Combinatorial optimization over two random point sets. Séminaire de Probabilités XLV, 483-535. Lecture Notes in Math. 2078, Springer, Cham, 2013.
• [7] Baum, L. E. and Katz, M.: Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, (1965), 108-123.
• [8] Bobkov, S. and Ledoux, M.: One-dimensional empirical measures, order statistics and Kantorovich transport distances. To appear in the Memoirs of the Amer. Math. Soc.
• [9] Boissard, E. and Le Gouic, T.: On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Ann. Inst. Henri Poincaré Probab. Stat. 50, (2014), 539-563.
• [10] Dedecker, J. and Merlevède, F.: Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences. Bernoulli 23, (2017), 2083-2127.
• [11] Dolera, E. and Regazzini, E.: Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences. (2017) arXiv:1712.07361v2
• [12] Dereich, S., Scheutzow, M. and Schottstedt, R.: Constructive quantization: approximation by empirical measures. Ann. Inst. Henri Poincaré Probab. Stat. 49, (2013), 1183-1203.
• [13] Dobrić, V. and Yukich, J. E.: Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 8, (1995), 97-118.
• [14] Dudley, R. M.: The speed of mean Glivenko-Cantelli convergence. Ann. Math. Statist. 40, (1968), 40-50.
• [15] Èbralidze, S. S.: Inequalities for the probabilities of large deviations in terms of pseudomoments, (Russian). Teor. Verojatnost. i Primenen 16, (1971), 760-765.
• [16] Fournier, N. and Guillin, A.: On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Relat. Fields 162, (2015), 707-738.
• [17] Kloeckner, B.: Empirical measures: regularity is a counter curse to dimensionality. (2018) arXiv:1802.04038v1
• [18] Ledoux, M. and Talagrand, M.: Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23 Springer-Verlag, 1991, Berlin, xii + 480 pp.
• [19] Le Gouic, T.: Localisation de masse et espace de Wasserstein. PhD Thesis, Université Toulouse III-Paul Sabatier. (2013).
• [20] Marcinkiewicz, J. and Zygmund, A.: Sur les fonctions indépendantes. Fund. Math. 29, (1937), 60-90.
• [21] Pinelis, I.: Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22, (1994), 1679-1706.
• [22] Rosenthal, H. P.: On the subspaces of $L^p \, (p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8, (1970), 273-303.
• [23] Talagrand, M.: Matching random samples in many dimensions. Ann. Appl. Probab. 2, (1992), 846-856.