On the functional central limit theorem for stationary processes

Abstract

We give a sufficient condition for a stationary sequence of square-integrable and real-valued random variables to satisfy a Donsker-type invariance principle. This condition is similar to the L 1 -criterion of Gordin for the usual central limit theorem and provides invariance principles for α-mixing or β-mixing sequences as well as stationary Markov chains. In the latter case, we present an example of a non irreducible and non α-mixing chain to which our result applies.