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Epi-convergence of Sample Averages of a Random Lower Semi-continuous Functional Generated by a Markov Chain and Application to Stochastic Optimization

Arnab Sur(arnabsur2002***at***gmail.com)
John Birge(john.birge***at***chicagobooth.edu)

Abstract: 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.

Keywords: Sample average approximation method, Epigraphical convergence, Random lower semi-continuous function, Measure-preserving ergodic transformation, Harris recurrent Markov chain, Consistency, Asymptotic normality.

Category 1: Stochastic Programming

Citation: MOR-2020-092, The University of Chicago, March/2020

Download: [PDF]

Entry Submitted: 04/21/2020
Entry Accepted: 04/21/2020
Entry Last Modified: 04/21/2020

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