Open Access
2016 Convergence in density in finite time windows and the Skorohod representation
Hermann Thorisson
Electron. Commun. Probab. 21: 1-9 (2016). DOI: 10.1214/16-ECP4644


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 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 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.


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Hermann Thorisson. "Convergence in density in finite time windows and the Skorohod representation." Electron. Commun. Probab. 21 1 - 9, 2016.


Received: 19 October 2015; Accepted: 15 August 2016; Published: 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1348.60006
MathSciNet: MR3548775
Digital Object Identifier: 10.1214/16-ECP4644

Primary: 60B10
Secondary: 60G99

Keywords: convergence in density , Convergence in distribution , Skorohod representation , widening time window

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