On the Convergence of Random Convex Sets

Abstract

The study of the convergence of random convex sets is motivated by various applications in statistics [1, 2] probability [3, 4] as well as by many approximation problems in stochastic optimization, see for example [5, 6, 7, 8]. More specifically one seeks to find conditions to ensure the convergence in distribution, in probability or almost everywhere of the optimal value of a sequence of problems { P ν , ν ∈ ℵ} to the optimal value of a problem P when the constraints and the objective function of the problems P ν converge in some probabilistic sense to the constraints and the objective function of P. Here, we only consider the convex case, i.e. when P and the P ν can be expressed as the minimization of convex functionals (with values in the extended reals). More precisely, let (Ξ, F) be a measure space and {σ ν , ν ∈ ℵ} a sequence of probability measures “converging” to a probability measure σ, all defined on (Ξ, F). Let f be a functional with domain Ξ×E with values in ℛ ∪{ + ∞} where E is a finite dimensional Euclidean space.