Strong approximation results for the empirical process of stationary sequences

Abstract

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.

Published in at http://dx.doi.org/10.1214/12-AOP798 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Strong invariance principles for dependent random variables
• Some unbounded functions of intermittent maps for which the central limit theorem holds

Cited by in corpus (2)

• A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to B-Geometrically Ergodic Markov Chains
• Weak convergence of the empirical process of intermittent maps in L2 under long-range dependence