On parametric models for invariant probability measures

Let $x=(x_n:n \in N)$ be a sequence of random variables with values in a Polish space X. If x is exchangeable (stationary), then x is i.i.d. (ergodic) conditionally on some random probability measure p on B(x) (q on B($X^\infty$). In the exchangeable case, two necessary and sufficient conditions are given for x to be i.i.d. conditionally on t(p), where t is a given transformation. Essentially the same conditions apply when x is stationary, apart from "i.i.d." is replaced by "ergodic" and p by q. The basic difference between the exchangeable and the stationary case lies in the empirical measure to be used. Up to a proper choice of the latter, the two conditions work in a large class of invariant distributions for x. Finally, the set of ergodic probability measures is shown to be a Borel set, and almost sure weak convergence of a certain sequence of empirical measures is proved in the stationary case.