Moderate Deviations and Invariance Principles for Sample Average Approximations

We study moderate deviations and convergence rates for the optimal values and optimal solutions of sample average approximations. Firstly, we give an extension of the Delta method in large deviations. Then under Lipschitz continuity on the objective function, we establish a moderate deviation principle for the optimal value by the Delta method. When the objective function is twice continuously differentiable and the optimal solution of true optimization problem is unique, we obtain a moderate deviation principle for the optimal solution and a Cram\'er-type moderate deviation for the optimal value. Motivated by the Donsker invariance principle, we consider a functional form of stochastic programming problem and establish a Donsker invariance principle, a functional moderate deviation principle, and a Strassen invariance principle for the optimal value.