Convergence in Density in Finite Time Windows and the Skorohod Representation

TL;DR

This paper extends the Skorohod representation theorem to stochastic processes over expanding time windows, establishing conditions for convergence in density and almost sure convergence in the discrete metric.

Contribution

It generalizes the discrete-metric convergence theorem to processes in growing time windows, leading to a separability-based Skorohod representation.

Findings

Extended the discrete-metric theorem to stochastic processes in expanding windows

Proved a separability version of the Skorohod representation theorem

Established conditions for density convergence and almost sure convergence

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.