Content:
We prove an Ergodic Theorem for random lsc (lower semicontinuous) functions. The key technique is `scalarization'.
Solution procedures for stochastic programming problems, statistical estimation problems (constrained or not), stochastic optimal control problems and other stochastic optimization problems often rely on sampling. The justification for such an approach passes through `consistency.' A comprehensive, satisfying and powerful technique is to obtain the consistency of the optimal solutions, statistical estimators, controls, etc., as a consequence of the consistency of the stochastic optimization problems themselves. And to do this, as explained in Section 2, one can appeal to the ergodicity properties of random lsc (lower semicontinuous) functions set forth in this paper.
A streamlined version of this basic ergodic theorem coud be formulated as follows: Let (X,d) be a Polish space and (Z, S, P) a probability space.
Theorem. Let f be a random lsc function defined on ZxX, v: Z -> Z an ergodic measure preserving transformation. Then, under a integrability condition for the inf-function z -> inf f(z,.), the empirical means of f(v^k(z), .) epi-converge to Ef, the expectation of f.
January 2000