  According to the Dudley-Wichura extension of the Skorohod representation
theorem, convergence in distribution to a limit in a separable set is
equivalent to the existence of a coupling with elements converging a.s. in the
metric. A density analogue of this theorem says that a sequence of probability
densities on a general measurable space has a probability density as a
pointwise lower limit if and only if there exists a coupling with elements
converging a.s. in the discrete metric. In this paper the discrete-metric
theorem is extended to stochastic processes considered in a widening time
window. The extension is then used to prove the separability version of the
Skorohod representation theorem.
