Set Estimation from Projected Multidimensional Random Variables with Application to a Discrete-Time Skorokhod Problem

This paper deals with sufficient conditions on the distribution of the random variable $H$, in the model $X =\Pi_C(H)$, for the convex hull $\widehat C_N$ of $N$ independent copies of $X$ to be a consistent estimator of the convex body $C$ with a rate of convergence. The convergence of $\widehat C_N$ is established for the Hausdorff distance under a uniform condition on the distribution of $H$, but also in a pointwise sense under a less demanding condition. Some of these convergence results on $\widehat C_N$ are applied to the estimation of the time-dependent constraint set involved in a discrete-time Skorokhod problem.