Uniform Convergence in Extended Probability of Sub-Gradients of Convex Functions

It is well known that if a sequence of stochastic convex functions on $\mathbb{R}^{d}$ converges in probability point-wise to some non-stochastic function then the limit function is convex and the convergence is uniform on compact sets; see Andersen and Gill (1982) and Pollard (1991). In the present paper, I establish that if the limiting function is differentiable then any sequence of measurable sub-gradients of the stochastic convex functions converges in extended probability to the gradient of the limit function uniformly on compact sets.