  Let $(X_i)_{i\in\Z}$ be a regular stationary process for a given filtration.
The weak invariance principle holds under the condition
$\sum_{i\in\Z}\|P_0(X_i)\|_2<\infty$ (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)).
