Arginf-sets of multivariate cadlag processes and their convergence in hyperspace topologies

D. Ferger
Theory of Stochastic Processes
Vol.20 (36), no.2, 2015, pp.13-41
Let X_n, n ∈ N, be a sequence of stochastic processes with trajectories in the multivariate Skorokhod-space D(R^d). If A(X_n) denotes the set of all infimizing points of X_n, then A(X_n) is shown to be a random closed set, i.e. a random variable in the hyperspace F, which consists of all closed subsets of R^d. We prove that if X_n converges to X in D(R^d) in probability, almost surely or in distribution, then A(X_n) converges in the analogous manner to A(X) in F endowed with appropriate hyperspace topologies. Our results immediately yield continuous mapping theorems for measurable selections ξ_n ∈ A(X_n). Here we do not require that A(X) is a singleton as it is usually assumed in the literature. In particular it turns out that ξ_n converges in distribution to a Choquet capacity, namely the capacity functional of A(X). In fact, this motivates us to extend the classical concept of weak convergence. In statistical applications it facilitates the construction of confidence regions based on M-estimators even in the case that the involved limit process has no longer an a.s. unique infimizer as it was necessary so far.