On coupling and convergence in density and in distribution

According to Dudley’s extension of the Skorohod representation theo-rem, convergence in distribution on a separable metric space is equiv-alent 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 lower pointwise limit if and only if there exists a cou-pling with elements converging a.s. in the discrete topology. In this paper the latter result is extended to discrete-topology convergence of stochastic processes in a widening time-window. An elementary ver-sion of that result is then used to prove the Skorohod-Dudley theorem. 1 Introduction and statement of results The aim of this paper is to present a coupling characterization of finite-win-dows density convergence of a sequence of stochastic processes (Theorem 1), and to use an elementary version of that result (Corollary 1) to prove Dudley’