Strong approximation results for the empirical process of stationary sequences. (English) Zbl 1284.60069

The authors prove a strong invariance principle for the sequential empirical process, i.e., an almost sure approximation of the centered and rescaled empirical distribution function by a Gaussian process. As corollaries, weak convergence of this process and a law of the iterated logarithm are obtained. The empirical distribution function is associated to a class of stationary, dependent sequences of random variables, using a dependence condition that is similar in spirit to absolute regularity, but it only involves indicators of half lines. No condition on the differentiability of the distribution function is needed and the assumption on the decay of the dependence coefficient is mild. The result can be applied to the sequence of iteration of an intermittend map, which has a neutral fixed point at the origin.


60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
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