Epi-convergence almost surely, in probability and in distribution

The paper deals with an epi-convergence of random real functions defined on a topological space. We follow the idea due to Vogel (1994) to split the epi-convergence into the lower semicontinuous approximation and the epi-upper approximation and localize them onto a given set. The approximations are shown to be connected to the miss- resp. hit-part of the ordinary Fell topology on sets. We introduce two procedures, called “localization”, separately for the miss-topology and the hit-topology on sets. Localization of the miss- resp. hit-part of the Fell topology on sets allows us to give a suggestion how to define the approximations in probability and in distribution. It is shown in the paper that in case of the finite-dimensional Euclidean space, the suggested approximations in probability coincide with the definition from Vogel and Lachout (2003).