Electronic Communications in Probability

Convergence in density in finite time windows and the Skorohod representation

Hermann Thorisson

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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.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 63, 9 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1473685927

Digital Object Identifier
doi:10.1214/16-ECP4644

Mathematical Reviews number (MathSciNet)
MR3548775

Zentralblatt MATH identifier
1348.60006

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60G99: None of the above, but in this section

Keywords
Skorohod representation convergence in distribution convergence in density widening time window

Rights
Creative Commons Attribution 4.0 International License.

Citation

Thorisson, Hermann. Convergence in density in finite time windows and the Skorohod representation. Electron. Commun. Probab. 21 (2016), paper no. 63, 9 pp. doi:10.1214/16-ECP4644. https://projecteuclid.org/euclid.ecp/1473685927


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References

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