Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. The proof we use is simple and of interest even when other proofs are available.
"On Vague Convergence of Stochastic Processes." Ann. Probab. 3 (6) 1014 - 1022, December, 1975. https://doi.org/10.1214/aop/1176996227