In this paper we extend our asymptotic results [l], [2] to a more general setting. Let R be a space of functions w (. ) on -00 < t < 00 with values in a Polish space X. We assume R consists of functions with discontinuities only of the first kind, normalized to be right continuous, and with convergence induced by the Skorohod topology on bounded intervals. In this case, R is itself a Polish space. Denote by 0: the corresponding space of functions on [t, 00) with values in X.We denote by S S the a-field in R generated by w (a) for s 5 a 5 t. We denote the translation map on R by Or, i.e., (O,w)(s) = w (s + t).Let Po,, be a Markov process on a,' starting from x E X satisfying the hypothesis that the mapping x +PO,, is weakly continuous (which implies the Feller property for the process PO,,).Let w E R and, for each t >0, define wt by
w r ( s ) = w ( s ) ,
O S s < t ,wt(s + t ) = w f ( s ) for all s E (-00, 00).
Now, (6,wt)(T) = wt(s + T ) , 0 5 s 5 t, and for any set A c R we defineIf we let &(R) denote the space of stationary processes on R, we see that, for each w E R and each t > 0, I?',,( a ) E &(R). This is because, for any a > 0,