Weak convergence and tightness of probability measures in an abstract Skorohod space

Abstract

In this article, we introduce the space $D([0,1];D)$ of functions defined on $[0,1]$ with values in the Skorohod space $D$, which are right-continuous and have left limits with respect to the $J_1$ topology. This space is equipped with the Skorohod-type distance introduced in Whitt (1980). Following the classical approach of Billingsley (1968, 1999), we give several criteria for tightness of probability measures on this space, by characterizing the relatively compact subsets of this space. In particular, one of these criteria has been used in the recent article Balan and Saidani (2018) for proving the existence of a $D$-valued $\alpha$-stable L\'evy motion. Finally, we give a criterion for weak convergence of random elements in $D([0,1];D)$, and a criterion for the existence of a process with sample paths in $D([0,1];D)$ based on its finite-dimensional distributions.