I've read something about the splitting method (Asmussen: Stochastic Simulation, p. 147). There it's stated that:
If $X,Y$ are independet random variables, $Y_s$ are samples of $Y$, then $\frac{1}{S} \sum\limits_{s=1}^S \phi(X,Y_s)$ is a Monte Carlo estimate of $E[\phi(X,Y)|X]$.
So I think this means $P$-a.s.-convergence of the estimator to this conditional expectation. Does anyone know how to prove this or could give a hint for further reading?
Any help would be much appreciated. Thanks a lot.