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

Jérôme Dedecker and Florence Merlevède

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1489737634

Digital Object Identifier
doi:10.3150/16-BEJ805

Mathematical Reviews number (MathSciNet)
MR3624887

Zentralblatt MATH identifier
06714328

Keywords
central limit theorem empirical process intermittency moments inequalities stationary sequences Wasserstein distance

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


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