Translator Disclaimer
2014 Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$
Jérôme Dedecker, Emmanuel Rio, Florence Merlevède
Author Affiliations +
Electron. J. Probab. 19: 1-56 (2014). DOI: 10.1214/EJP.v19-2658

Abstract

We prove a strong approximation result with rates for the empirical process associated to an absolutely regular stationary sequence of random variables with values in ${\mathbb R}^d$. As soon as the absolute regular coefficients of the sequence decrease more rapidly than $n^{1-p} $ for some $p \in ]2,3]$, we show that the error of approximation between the empirical process and a two-parameter Gaussian process is of order $n^{1/p} (\log n)^{\lambda(d)}$ for some positive $\lambda(d)$ depending on $d$, both in ${\mathbb L}^1$ and almost surely. The power of $n$ being independent of the dimension, our results are even new in the independent setting, and improve earlier results. In addition, for absolutely regular sequences, we show that the rate of approximation is optimal up to the logarithmic term.

Citation

Download Citation

Jérôme Dedecker. Emmanuel Rio. Florence Merlevède. "Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$." Electron. J. Probab. 19 1 - 56, 2014. https://doi.org/10.1214/EJP.v19-2658

Information

Accepted: 14 January 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1293.60043
MathSciNet: MR3164762
Digital Object Identifier: 10.1214/EJP.v19-2658

Subjects:
Primary: 60F17
Secondary: 60G10

JOURNAL ARTICLE
56 PAGES


SHARE
Vol.19 • 2014
Back to Top