Kybernetika 39 no. 1, 75-98, 2003

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

Silvia Vogel and Petr Lachout

Abstract:

Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are 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.