Convergence in density in finite time windows and the Skorohod representation

Abstract

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. The paper concludes with an application to Markov chains.