Convergence in Distribution of Conditional Expectations

Abstract

Suppose the random variables $(X^N, Y^N)$ on the probability space $(\Omega^N, \mathscr{F}^N, P^N)$ converge in distribution to the pair $(X, Y)$ on $(\Omega, \mathscr{F}, P)$, as $N \rightarrow \infty$. This paper seeks conditions which imply convergence in distribution of the conditional expectations $E^{P^N}\{F(X^N)\mid Y^N\}$ to $E^P\{F(X)\mid Y\}$, 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 $E^{P^N}\{F(X^N)\mid Y^N\}$ (under the original distributions) to $E^P\{F(X)\mid Y\}$ 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.