## The Annals of Probability

### Central limit theorems for empirical transportation cost in general dimension

#### Abstract

We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on $\mathbb{R}^{d}$, with $d\geq1$. We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 926-951.

Dates
Revised: March 2018
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.aop/1551171641

Digital Object Identifier
doi:10.1214/18-AOP1275

Mathematical Reviews number (MathSciNet)
MR3916938

Zentralblatt MATH identifier
07053560

#### Citation

del Barrio, Eustasio; Loubes, Jean-Michel. Central limit theorems for empirical transportation cost in general dimension. Ann. Probab. 47 (2019), no. 2, 926--951. doi:10.1214/18-AOP1275. https://projecteuclid.org/euclid.aop/1551171641

#### References

• [1] Ajtai, M., Komlós, J. and Tusnády, G. (1984). On optimal matchings. Combinatorica 4 259–264.
• [2] Ambrosio, L., Stra, F. and Trevisan, D. (2016). A PDE approach to a 2-dimensional matching problem. Preprint. Available at arXiv:1611.04960.
• [3] Bobkov, S. and Ledoux, M. (2016). One-dimensional empirical measures, order statistics and Kantorovich transport distances. Memoirs of the AMS. To appear.
• [4] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
• [5] Cuesta-Albertos, J. A., Matrán, C. and Tuero-Dí az, A. (1997). Optimal transportation plans and convergence in distribution. J. Multivariate Anal. 60 72–83.
• [6] del Barrio, E., Giné, E. and Matrán, C. (1999). Central limit theorems for the Wasserstein distance between the empirical and the true distributions. Ann. Probab. 27 1009–1071.
• [7] del Barrio, E., Giné, E. and Utzet, F. (2005). Asymptotics for $L_{2}$ functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances. Bernoulli 11 131–189.
• [8] del Barrio, E. and Matrán, C. (2013). Rates of convergence for partial mass problems. Probab. Theory Related Fields 155 521–542.
• [9] Dobrić, V. and Yukich, J. E. (1995). Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 8 97–118.
• [10] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL.
• [11] Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 707–738.
• [12] Gangbo, W. and McCann, R. J. (1996). The geometry of optimal transportation. Acta Math. 177 113–161.
• [13] Heinich, H. and Lootgieter, J.-C. (1996). Convergence des fonctions monotones. C. R. Acad. Sci. Paris Sér. I Math. 322 869–874.
• [14] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems, Vols. I, II. Probability and Its Applications (New York). Springer, New York.
• [15] Rippl, T., Munk, A. and Sturm, A. (2016). Limit laws of the empirical Wasserstein distance: Gaussian distributions. J. Multivariate Anal. 151 90–109.
• [16] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
• [17] Rockafellar, R. T. and Wets, R. J. (1998). Variational Analysis. Springer, Berlin.
• [18] Sommerfeld, M. and Munk, A. (2018). Inference for empirical Wasserstein distances on finite spaces. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 219–238.
• [19] Talagrand, M. (1992). Matching random samples in many dimensions. Ann. Appl. Probab. 2 846–856.
• [20] Talagrand, M. (1994). The transportation cost from the uniform measure to the empirical measure in dimension ${\ge}3$. Ann. Probab. 22 919–959.
• [21] Talagrand, M. and Yukich, J. E. (1993). The integrability of the square exponential transportation cost. Ann. Appl. Probab. 3 1100–1111.
• [22] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
• [23] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.