A common approach to solve stochastic optimization problems with expectations is to replace the expectations by its sample averages. Large sample asymptotic properties of this approximation are well studied when the sample is i.i.d. In many cases, however, i.i.d. samples are not readily available. On the contrary, one can generate a Harris recurrent Markov chain with stationary distribution using Markov chain Monte Carlo (MCMC). We call it Markov chain sample average approximation (MCSAA) when the true average is replaced by the sample averages associated to a general state space Markov chain with stationary distribution. In this article, we study the large sample properties of MCSAA estimators associated to a random convex function and also construct probabilistic (exponential) error bounds using large deviation principle via weak convergence.
MAPR-D-20-00305, The University of Chicago, August/2020