Moderate Deviations for Stochastic Variational Inequalities

Stochastic variational inequalities (SVIs) have been used widely in modelling various optimization and equilibrium problems subject to data uncertainty. The sample average approximation (SAA) solution is an asymptotically consistent point estimator for the true solution to a stochastic variational inequality. Some central limit results and large deviation estimates for the SAA solution have been obtained. The purpose of this paper is to study the convergences in regimes of moderate deviations for the SAA solution. Using the delta method and the exponential approximation, we establish some results on moderate deviations. We apply the results to the hypotheses testing for the SVIs, and prove that the rejection region constructed by the central limit theorem has the probability of the type II error with exponential decay speed. We also give some simulations and numerical results for the tail probabilities.