(Y, ) is a stationary dilation of X.

Without loss of generality, one can assume that Yt is purely nondeterministic. If not, let Yt = Rt + St be the Wold decomposition of Yt and notice that (H-S( )) = {0} since Xt is purely nondeterministic. Let : HY ( ) H R ( ) be the orthogonal projection of HY( ) onto HR( ). Then (Rt, ) and ( , ) are stationary dilations of Xt and , respectively. One can use ( t) and Rt instead of t and Yt.