Convergence results for conditional expectations

Abstract

Let E,F be two Polish spaces and [X_n,Y_n],[X,Y] random variables with values in E×F (not necessarily defined on the same probability space). We show some conditions which are sufficient in order to assure that, for each bounded continuous function f on E×F, the conditional expectation of f(X_n,Y_n) given Y_n converges in distribution to the conditional expectation of f(X,Y) given Y.