CONVERGENCE IN DISTRIBUTION OF CONDITIONAL EXPECTATIONS

Suppose the random variables (X(N), Y(N)) on the probability space (OMEGA(N), F(N), P(N)) converge in distribution to the pair (X, Y) on (OMEGA, F, P), as N --> infinity. This paper seeks conditions which imply convergence in distribution of the conditional expectations E(PN){F(X(N))\Y(N)} to E(P){F(X)\Y}, for all bounded continuous functions F. An absolutely continuous change of probability measure is made from P(N) to a measure Q(N) under which X(N) and Y(N) are independent. The Radon-Nikodym derivative dP(N)/dQ(N) is denoted by L(N). Similarly, an absolutely continuous change of measure from P to Q is made, with Radon-Nikodym derivative dP/dQ = L. If the Q(N)-distribution of (X(N), Y(N), L(N)) converges weakly to the Q-distribution of (X, Y, L), convergence in distribution of E(PN)(F(X(N)\Y(N)) (under the original distributions) to E(P){F(X)\Y} follows. Conditions of a uniform equicontinuity nature on the L(N) are presented which imply the required convergence. Finally, an example is given, where convergence of the conditional expectations can be shown quite easily.