Convergence of Subdifferentials Under Strong Stochastic Convexity

We show that if a sequence of random functions satisfies strong stochastic convexity with respect to a parameter, and if the sequence converges pointwise with probability one, then any sequence of elements extracted from the subdifferentials of the functions in the sequence will converge to the subdifferential of the limiting function, again with probability one. This result holds with no differentiability assumption on the limiting function, and even if the limiting function is itself random. It thus extends earlier work, in particular results by Glynn and by Hu. One application is in proving an extended form of strong consistency for infinitesimal perturbation analysis (IPA) when suitable convexity properties hold.