On continuous convergence and epi-convergence of random functions - Part I: Theory and relations

Continuous convergence and epi-convergence of sequences of random functions axe crucial assumptions if mathematical programming problems axe approximated on the basis of estimates or via sampling. The paper investigates "almost surely" and "in probability" versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization.