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Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$


 
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1. Title Title of document Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$
 
2. Creator Author's name, affiliation, country Jérôme Dedecker; Université Paris Descartes; France
 
2. Creator Author's name, affiliation, country Emmanuel Rio; Université de Versailles; France
 
2. Creator Author's name, affiliation, country Florence Merlevède; Université Paris-Est Marne-la-Vallée; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Strong approximation, Kiefer process, empirical process, stationary sequences, absolutely regular sequences.
 
3. Subject Subject classification 60F17, 60G10
 
4. Description 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.

 
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2014-01-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2658
 
10. Identifier Digital Object Identifier 10.1214/EJP.v19-2658
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 19
 
12. Language English=en en
 
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