Convergence of Subdifferentials Under Strong Stochastic Convexity
证明，若随机函数序列满足强随机凸性且逐点几乎必然收敛，则其次微分序列也几乎必然收敛到极限函数的次微分，无需极限函数可微，适用于扰动分析的一致性证明。
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.