On a strong graphical law of large numbers for random semicontinuous mappings

In the paper we establish a strong graphical law of large numbers (LLN) for random outer semicontinuous mappings, providing conditions when graphs of sample average mappings converge to the graph of the expectation mapping with probability one. This result extends a known LLN for compact valued random sets to random uniformly bounded (by an integrable function) set valued mappings. We give also an equivalent formulation for the graphical LLN by means of some fattened mappings. The study is motivated by applications of the set convergence and the graphical LLN in stochastic variational analysis, including approximation and solution of stochastic generalized equations, stochastic variational inequalities and stochastic optimization problems. The nature of these applications consists in sample average approximation of the inclusion mappings, application of the graphical LLN and obtaining from here a graphical approximation of the set of solutions. Bibliogr. 23.