Electronic Communications in Probability

A Skorohod representation theorem without separability

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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Abstract

Let $(S,d)$ be a metric space, $\mathcal{G}$ a $\sigma$-field on $S$ and $(\mu_n:n\geq 0)$ a sequence of probabilities on $\mathcal{G}$. Suppose $\mathcal{G}$ countably generated, the map $(x,y)\mapsto d(x,y)$ measurable with respect to $\mathcal{G}\otimes\mathcal{G}$, and $\mu_n$ perfect for $n>0$. Say that $(\mu_n)$ has a Skorohod representation if, on some probability space, there are random variables $X_n$ such that<br />\begin{equation*}<br />X_n\sim\mu_n\text{ for all }n\geq 0\quad\text{and}\quad d(X_n,X_0)\overset{P}\longrightarrow 0.<br />\end{equation*}<br />It is shown that $(\mu_n)$ has a Skorohod representation if and only if<br />\begin{equation*}<br />\lim_n\,\sup_f\,\left|\mu_n(f)-\mu_0(f)\right|=0,<br />\end{equation*}<br />where $\sup$ is over those $f:S\rightarrow [-1,1]$ which are $\mathcal{G}$-universally measurable and satisfy $\left|f(x)-f(y)\right|\leq 1\wedge d(x,y)$. An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if $\mu_0$ fails to be $d$-separable. Some possible applications are given as well.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 80, 12 pp.

Dates
Accepted: 18 October 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315619

Digital Object Identifier
doi:10.1214/ECP.v18-2793

Mathematical Reviews number (MathSciNet)
MR3125256

Zentralblatt MATH identifier
1308.60007

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
Convergence of probability measures Perfect probability measure Separable probability measure Skorohod representation theorem Uniform distance

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

Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. A Skorohod representation theorem without separability. Electron. Commun. Probab. 18 (2013), paper no. 80, 12 pp. doi:10.1214/ECP.v18-2793. https://projecteuclid.org/euclid.ecp/1465315619


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References

  • Basse-O'Connor, A. and Rosinski, J.: On the uniform convergence of random series in Skorohod space and representations of cadlag infinitely divisible processes. Ann. Probab., (2012), to appear.
  • 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.
  • Blackwell, David. On a class of probability spaces. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, pp. 1–6. University of California Press, Berkeley and Los Angeles, 1956.
  • 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
  • Jain, Naresh C.; Monrad, Ditlev. Gaussian measures in $B_{p}$. Ann. Probab. 11 (1983), no. 1, 46–57.
  • 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
  • Ramachandran, D.; Rüschendorf, L. A general duality theorem for marginal problems. Probab. Theory Related Fields 101 (1995), no. 3, 311–319.
  • 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.
  • 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
  • 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.