The purpose of this article is to establish epigraphical convergence of the sample averages of a random lower semi-continuous functional associated with a Harris recurrent Markov chain with stationary distribution $\pi$. Sample averages associated with an ergodic Markov chain with stationary probability distribution will epigraphically converge from $\pi$-almost all starting points. The property of Harris recurrence allows us to replace “almost all” by “all”, which is potentially important when running Markov chain Monte Carlo algorithms. That result on epi-convergence is then applied to establish the consistency of the optimal solutions and optimal value of a stochastic optimization problem involving expectation functional of the form $ E_{\pi}[f(x,\xi)].$ Moreover, we develop asymptotic normality of the statistical estimator of the optimal value using a Markov chain central limit theorem.
Citation
MOR-2020-092, The University of Chicago, March/2020