Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 63, 9 pp.
Convergence in density in finite time windows and the Skorohod representation
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.
Electron. Commun. Probab., Volume 21 (2016), paper no. 63, 9 pp.
Received: 19 October 2015
Accepted: 15 August 2016
First available in Project Euclid: 12 September 2016
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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