  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.
