Convergence of conditional expectations given the random variables of a Skorohod representation (Q1807913)

It is shown that under some technical condition on probability distributions \(P_n\Rightarrow P\) the Skorokhod representations \(X_n, X\) have the following interesting property: For any \(L^2\) random variable \(Z\) on \([0,1]\) the conditional expectations \(E(Z\mid {\mathcal F}^{X_n})\) converge in \(L^2\) to \(E(Z\mid {\mathcal F}^X)\). The proof is carried out by a detailed analysis of Billingsley's original proof of the Skorokhod representation together with martingale convergence arguments and estimates of conditional expectations in terms of sub-sigma-algebras. In an example it is shown that some additional condition is necessary for the main theorem to hold.