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

D. Ferger

Teoriâ slučajnyh processov, Tome 20 (2015) no. 2, pp. 13-41

Résumé

Let $X_n, n \in \mathbb{N}$, be a sequence of stochastic processes with trajectories in the multivariate Skorokhod-space $D(\mathbb{R}^d)$. 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 $\mathcal{F}$, which consists of all closed subsets of $\mathbb{R}^d$. We prove that if $X_n$ converges to $X$ in $D(\mathbb{R}^d)$ in probability, almost surely or in distribution, then $A(X_n)$ converges in the analogous manner to $A(X)$ in $\mathcal{F}$ endowed with appropriate hyperspace topologies. Our results immediately yield continuous mapping theorems for measurable selections $\xi_n \in A(X_n)$. 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 $\xi_n$ 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.

Détail

Keywords: Multivariate Skorokhod-space, weak convergence, sets of infimizing points, random closed sets, hyperspace topologies, continuous mapping theorems.
Mots-clés : epi-convergence, Choquet capacity