The Annals of Probability

Strong approximation results for the empirical process of stationary sequences

Jérôme Dedecker, Florence Merlevède, and Emmanuel Rio

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We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than $n^{-1-\delta}$ for some positive $\delta$. This shows that our conditions are in some sense optimal.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3658-3696.

First available in Project Euclid: 12 September 2013

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G10: Stationary processes 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

Strong approximation Kiefer process stationary sequences intermittent maps weak dependence


Dedecker, Jérôme; Merlevède, Florence; Rio, Emmanuel. Strong approximation results for the empirical process of stationary sequences. Ann. Probab. 41 (2013), no. 5, 3658--3696. doi:10.1214/12-AOP798.

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