Consistency of statistical estimators of solutions to stochastic optimization problems

Epi-Convergence of Expectation Functions under Varying Measures and Integrands

“…Theorem 3.7 strengthens the results of [29,28] by considering approximating functions f ν rather than a fixed integrand f . Specifically, [28, Theorem 1] allows X to be a Hausdorff topological space and confirms epi-convergence of E P ν f to E P f under the assumptions that f is lsc and f (•, x) is usc for each x.…”

“…As in [17], the main assumptions are expressed in terms of Pasch-Hausdorff envelopes. Theorem 3.7 establishes epi-convergence in the setting with weakly converging probabilities and extends [28,Theorem 1] to varying integrands. (While we restrict the attention to metric spaces, [28] deals with Hausdorff topological spaces; see below for a detailed comparison.)…”

“…Theorem 3.7 establishes epi-convergence in the setting with weakly converging probabilities and extends [28,Theorem 1] to varying integrands. (While we restrict the attention to metric spaces, [28] deals with Hausdorff topological spaces; see below for a detailed comparison.) Here, we dispense of Pasch-Hausdorff envelopes and instead rely on Fatou's lemmas for weakly convergent probabilities [12,9,10].…”

“…In contrast to [17], [28], and earlier studies, we also bring forth bounds on lower epi-limits, which emerge as useful in their own right as demonstrated in Section 4. These bounds appear as Lemma 3.1 and Theorem 3.3.…”

“…Going beyond empirical distributions, [18,7] establish epi-convergence of expectation functions for ergodic processes. Recently, [28] reexamines epi-convergence under weakly converging probabilities as in [29], but now in the general setting of Hausdorff topological spaces. Without making explicit assumptions on the sequence of probabilities, but instead expressing assumptions in terms of Pasch-Hausdorff envelopes, [17] develops a general framework for establishing epi-convergence of expectation functions defined on separable metric spaces.…”