## 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

#### 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