On a uniform law of large numbers for random sets and subdifferentials of random functions

In this paper, we strengthen the convergence in a uniform law of large numbers for random upper semicontinuous multifunctions of Shapiro and Xu. The proof is based on an abstract law of large numbers in a metric space endowed with a convex combination operation. Convergence in the Hausdorff metric is obtained, whereas the original result presented a weakened form of convergence of excess functionals. As a consequence, another law of large numbers for subdifferentials of random functions is improved as well.