LIMITS OF CONDITIONAL EXPECTATIONS

If (X(N), Y(N)) on a probability space (OMEGA(N), F(N), P(N)) converge in distribution to (X, Y) on (OMEGA, F, P), it is not necessarily true that the conditional expectations E(PN){F(X(N))\Y(N)} converge in distribution to E(P){F(X)\Y}, even for bounded, continuous functions F. The limits of the conditional expectations can be determined if it is possible to make an absolutely continuous change of probability measure, from p(N) to Q(N), so that, under Q(N), X(N) and Y(N) are independent. The aim of this paper is to extend this technique to the case where P(N) is ''not quite'' absolutely continuous with respect to this probability measure Q(N); but where there is a subset OMEGA1N of OMEGA(N) such that P(N) restricted to OMEGA1N is absolutely continuous with respect to Q(N), and such that P(N)(OMEGA1N) --> 1 as N --> infinity. The convergence results for the conditional expectations are shown to be the same as when the absolutely continuous change of measure can be made directly-the conditional expectations converge, but not necessarily to E(P){F(X)\Y}. A filtering application is examined, where a signal process X and observation Y satisfying a pair of stochastic differential equations, are approximated by the solutions X(N) and Y(N) of a pair of stochastic difference equations. If our approximating model employs noise increments with a density which must have compact support, conditions are given which ensure the expected convergence of the conditional expectations.