A Skorohod representation theorem for uniform distance

Let $mu_n$ be a probability measure on the Borel $sigma$-field on $D[0,1]$ with respect to Skorohod distance, $ngeq 0$. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are $D[0,1]$-valued random variables $X_n$ such that $X_nsimmu_n$ for all $ngeq 0$ and $ ormX_n-X_0 ightarrow 0$ in probability, where $ ormcdot$ is the sup-norm. Such conditions do not require $mu_0$ separable under $ ormcdot$. Applications to exchangeable empirical processes and to pure jump processes are given as well.