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2015 Gluing lemmas and Skorohod representations
Patrizia Berti, Luca Pratelli, Pietro Rigo
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Electron. Commun. Probab. 20: 1-11 (2015). DOI: 10.1214/ECP.v20-3870

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.

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Patrizia Berti. Luca Pratelli. Pietro Rigo. "Gluing lemmas and Skorohod representations." Electron. Commun. Probab. 20 1 - 11, 2015. https://doi.org/10.1214/ECP.v20-3870

Information

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

zbMATH: 1330.60010
MathSciNet: MR3374303
Digital Object Identifier: 10.1214/ECP.v20-3870

Subjects:
Primary: 60B10
Secondary: 60A05 , 60A10

Keywords: Finitely additive probability , Gluing lemma , Skorohod representation theorem , Wasserstein distance

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