An invariance principle for non-adapted processes

We present an invariance principle for a non-adapted stationary sequence of random variables, conditional with respect to the sigma-algebra of invariant sets. It is a generalization of an invariance principle of Wu and Woodroofe (2004, Corollary 3) using a method introduced by Volny (2006). An example shows that the method cannot be used directly for a generalization of the invariance principle of Peligrad and Utev (2005).