## Electronic Journal of Probability

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

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 9, 56 pp.

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

https://projecteuclid.org/euclid.ejp/1465065651

Digital Object Identifier
doi:10.1214/EJP.v19-2658

Mathematical Reviews number (MathSciNet)
MR3164762

Zentralblatt MATH identifier
1293.60043

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G10: Stationary processes

Rights

#### Citation

Dedecker, Jérôme; Rio, Emmanuel; Merlevède, Florence. Strong approximation of the empirical distribution function for absolutely regular sequences in ${\mathbb R}^d$. Electron. J. Probab. 19 (2014), paper no. 9, 56 pp. doi:10.1214/EJP.v19-2658. https://projecteuclid.org/euclid.ejp/1465065651

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