Sample Average Approximation Over Function Spaces: Statistical Consistency and Rate of Convergence

This paper considers sample average approximation (SAA) of a general class of stochastic optimization problems over a function space constraint set and driven by 'regulated' Gaussian processes. We estab-lish statistical consistency by proving equiconvergence of the SAA estimator via a sophisticated sample complexity result. Next, recognizing that implementation over such infinite-dimensional spaces is possible only if numerical optimization is performed over a finite-dimensional subspace of the constraint set, and if sample paths of the driving process can be generated over a finite grid, we identify the decay rate of the SAA estimator's expected optimality gap as a function of the optimization error, Monte Carlo sampling error, path generation approximation error, and subspace projection error.