## Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 2083-2127.

### Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences

#### Abstract

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary $\alpha$-dependent sequences. We prove some moments inequalities of order $p$ for any $p\geq1$, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well-known von Bahr–Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identically distributed random variables.

#### Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 2083-2127.

Dates
Revised: October 2015
First available in Project Euclid: 17 March 2017

https://projecteuclid.org/euclid.bj/1489737634

Digital Object Identifier
doi:10.3150/16-BEJ805

Mathematical Reviews number (MathSciNet)
MR3624887

Zentralblatt MATH identifier
06714328

#### Citation

Dedecker, Jérôme; Merlevède, Florence. Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences. Bernoulli 23 (2017), no. 3, 2083--2127. doi:10.3150/16-BEJ805. https://projecteuclid.org/euclid.bj/1489737634

#### References

• [1] Bobkov, S. and Ledoux, M. (2014). One-dimensional empirical measures, order statistics and Kantorovich transport distances. Preprint.
• [2] Bradley, R.C. (1997). On quantiles and the central limit question for strongly mixing sequences. J. Theoret. Probab. 10 507–555.
• [3] Cuny, C. (2016). Limit theorems under the Maxwell–Woodroofe condition in Banach spaces. Ann. Probab. To appear. Available at arXiv:1403.0772.
• [4] de Acosta, A., Araujo, A. and Giné, E. (1978). On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces. In Probability on Banach Spaces. Adv. Probab. Related Topics 4 1–68. New York: Dekker.
• [5] Dede, S. (2009). An empirical central limit theorem in ${\mathbf{L}}^{1}$ for stationary sequences. Stochastic Process. Appl. 119 3494–3515.
• [6] Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications. Stochastic Process. Appl. 106 63–80.
• [7] Dedecker, J., Gouëzel, S. and Merlevède, F. (2010). Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 46 796–821.
• [8] Dedecker, J. and Merlevède, F. (2015). Moment bounds for dependent sequences in smooth Banach spaces. Stochastic Process. Appl. 125 3401–3429.
• [9] Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Probab. Stat. 36 1–34.
• [10] 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.
• [11] Doukhan, P., Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Stat. 30 63–82.
• [12] Èbralidze, Š.S. (1971). Inequalities for the probabilities of large deviations in terms of pseudomoments. Teor. Verojatnost. i Primenen. 16 760–765.
• [13] Esseen, C. and Janson, S. (1985). On moment conditions for normed sums of independent variables and martingale differences. Stochastic Process. Appl. 19 173–182.
• [14] 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.
• [15] Gordin, M.I. (1973). Abstracts of communication, T.1:A-K. In International Conference on Probability Theory. Vilnius.
• [16] Gouëzel, S. (2004). Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 82–122.
• [17] Hannan, E.J. (1973). Central limit theorems for time series regression. Z. Wahrsch. Verw. Gebiete 26 157–170.
• [18] Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Berlin: Springer.
• [19] Jain, N.C. (1977). Central limit theorem and related questions in Banach space. In Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) 55–65. Providence, RI: Amer. Math. Soc.
• [20] Liverani, C., Saussol, B. and Vaienti, S. (1999). A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 671–685.
• [21] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
• [22] Merlevède, F. and Peligrad, M. (2013). Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples. Ann. Probab. 41 914–960.
• [23] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Berlin: Springer.
• [24] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
• [25] Tucker, H.G. (1959). A generalization of the Glivenko–Cantelli theorem. Ann. Math. Statist. 30 828–830.
• [26] von Bahr, B. and Esseen, C. (1965). Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq2$. Ann. Math. Statist 36 299–303.