Electronic Communications in Probability

Gluing lemmas and Skorohod representations

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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Abstract

Let $(\mathcal{X},\mathcal{E})$, $(\mathcal{Y},\mathcal{F})$ and $(\mathcal{Z},\mathcal{G})$ be measurable spaces. Suppose we are given two probability measures $\gamma$ and $\tau$, with $\gamma$ defined on $(\mathcal{X}\times\mathcal{Y},\mathcal{E}\otimes\mathcal{F}$ and $\tau$ on $(\mathcal{X}\times\mathcal{Z},\mathcal{E}\otimes\mathcal{G})$. Conditions for the existence of random variables $X,Y,Z$, defined on the same probability space $(\Omega,\mathcal{A},P)$ and satisfying

$$(X,Y)\sim\gamma\,\text{ and }\,(X,Z)\sim\tau,$$

are given. The probability $P$ may be finitely additive or $\sigma$-additive. As an application, a version of Skorohod representation theorem is proved. Such a version does not require separability of the limit probability law, and answers (in a finitely additive setting) a question raised in preceding works.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 53, 11 pp.

Dates
Accepted: 21 July 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320980

Digital Object Identifier
doi:10.1214/ECP.v20-3870

Mathematical Reviews number (MathSciNet)
MR3374303

Zentralblatt MATH identifier
1330.60010

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

Keywords
Finitely additive probability Gluing lemma Skorohod representation theorem Wasserstein distance

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Gluing lemmas and Skorohod representations. Electron. Commun. Probab. 20 (2015), paper no. 53, 11 pp. doi:10.1214/ECP.v20-3870. https://projecteuclid.org/euclid.ecp/1465320980


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