Electronic Communications in Probability

Gluing lemmas and Skorohod representations

Patrizia Berti, Luca Pratelli, and Pietro Rigo

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Finitely additive probability Gluing lemma Skorohod representation theorem Wasserstein distance

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


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

Export citation


  • Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Skorohod representation on a given probability space. Probab. Theory Related Fields 137 (2007), no. 3-4, 277–288.
  • Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Skorohod representation theorem via disintegrations. Sankhya A 72 (2010), no. 1, 208–220.
  • Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. A Skorohod representation theorem for uniform distance. Probab. Theory Related Fields 150 (2011), no. 1-2, 321–335.
  • Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. A Skorohod representation theorem without separability. Electron. Commun. Probab. 18 (2013), no. 80, 12 pp.
  • Bhaskara Rao, K. P. S.; Bhaskara Rao, M. Theory of charges. A study of finitely additive measures. With a foreword by D. M. Stone. Pure and Applied Mathematics, 109. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. x+315 pp. ISBN: 0-12-095780-9
  • Dubins, Lester E. Paths of finitely additive Brownian motion need not be bizarre. S�minaire de Probabilit�s, XXXIII, 395–396, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Dudley, R. M. Distances of probability measures and random variables. Ann. Math. Statist 39 1968 1563–1572.
  • Dudley, R. M. Uniform central limit theorems. Cambridge Studies in Advanced Mathematics, 63. Cambridge University Press, Cambridge, 1999. xiv+436 pp. ISBN: 0-521-46102-2
  • Jakubowski, A. The almost sure Skorokhod representation for subsequences in nonmetric spaces. Teor. Veroyatnost. i Primenen. 42 (1997), no. 1, 209–216; translation in Theory Probab. Appl. 42 (1997), no. 1, 167–174 (1998)
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Karandikar, Rajeeva L. A general principle for limit theorems in finitely additive probability: the dependent case. J. Multivariate Anal. 24 (1988), no. 2, 189–206.
  • Ramachandran, D. The marginal problem in arbitrary product spaces. Distributions with fixed marginals and related topics (Seattle, WA, 1993), 260–272, IMS Lecture Notes Monogr. Ser., 28, Inst. Math. Statist., Hayward, CA, 1996.
  • Sethuraman, Jayaram. Some extensions of the Skorohod representation theorem. Special issue in memory of D. Basu. Sankhyā Ser. A 64 (2002), no. 3, part 2, 884–893.
  • Skorohod, A. V. Limit theorems for stochastic processes. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 289–319.
  • Thorisson, Hermann. Coupling methods in probability theory. Scand. J. Statist. 22 (1995), no. 2, 159–182.
  • Thorisson, Hermann. Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7
  • Thorisson, H.: Convergence in density in finite time windows and the Skorohod representation. phSubmitted, (2014).
  • van der Vaart, Aad W.; Wellner, Jon A. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. xvi+508 pp. ISBN: 0-387-94640-3
  • Villani, C�dric. Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. ISBN: 978-3-540-71049-3
  • Wichura, Michael J. On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Statist. 41 1970 284–291.