On Vague Convergence of Stochastic Processes

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.