On the Convergence in Distribution of Measurable Multifunctions (Random Sets) Normal Integrands, Stochastic Processes and Stochastic Infima

The concept of the distribution function of a closed-valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi-convergence in distribution of normal integrands and stochastic processes; in particular various compactness criteria are exhibited. The connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of Donsker's Theorem (Brownian motion as a limit of normalized random walks). We also suggest the potential application of the theory to the study of the convergence of stochastic infima.