On the convergence of sample approximations for stochastic programming problems with probabilistic criteria

We consider stochastic programming problems with probabilistic and quantile criteria. We describe a method for approximating these problems with a sample of realizations for random parameters. When we use this method, criterial functions of the problems are replaced with their sample estimates. We show the hypoconvergence of sample probability functions to its exact value that guarantees the convergence of approximations for the probability function maximization problem on a compact set with respect to both the value of the criterial function and the optimization strategy. We prove a theorem on the convergence of approximation for the quantile function minimization problem with respect to the value of the criterial function and the optimization strategy.