Independence of Four Projective Criteria for the Weak Invariance Principle

Let $(X_i{)}_{i\in \mathbb{Z}}$ be a regular stationary process for a given filtration. The weak invariance principle holds under the condition ${\sum }_{i\in \mathbb{Z}}\mathrm{\parallel }{P}_0(X_i){\mathrm{\parallel }}_2<\mathrm{\infty }$ (see Hannan (1979)}, Dedecker and Merlev`ede (2003), Deddecker, Merlev'ede and Voln'y (2007)). In this paper, we show that this criterion is independent of other known criteria: the martingale-coboundary decomposition of Gordin (see Gordin (1969, 1973)), the criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the condition of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade and Utev (2005), Voln'y (2006, 2007)).