Electronic Communications in Probability

On the existence of continuous processes with given one-dimensional distributions

Luca Pratelli and Pietro Rigo

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Abstract

Let $\mathcal{P} $ be the collection of Borel probability measures on $\mathbb{R} $, equipped with the weak topology, and let $\mu :[0,1]\rightarrow \mathcal{P} $ be a continuous map. Say that $\mu $ is presentable if $X_{t}\sim \mu _{t}$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu $ fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, $\mu $ is presentable if $\mu _{t}$ is supported by an interval (possibly, by a singleton) for all but countably many $t$. Secondly, assuming $\mu $ presentable, we investigate whether the quantile process $Q$ induced by $\mu $ has continuous paths. The latter is defined, on the probability space $((0,1),\mathcal{B} (0,1),\mbox{Lebesgue measure} )$, by \[ Q_{t}(\alpha )=\inf \, \bigl \{x\in \mathbb{R} :\mu _{t}(-\infty ,x]\ge \alpha \bigl \} \quad \quad \mbox{for all } t\in [0,1]\mbox{ and } \alpha \in (0,1). \] A few open problems are discussed as well.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 46, 9 pp.

Dates
Received: 12 June 2019
Accepted: 2 July 2019
First available in Project Euclid: 28 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1566957630

Digital Object Identifier
doi:10.1214/19-ECP255

Zentralblatt MATH identifier
07107310

Subjects
Primary: 60A05: Axioms; other general questions 60B10: Convergence of probability measures 60G05: Foundations of stochastic processes 60G17: Sample path properties

Keywords
Finite dimensional distributions process with continuous paths quantile process

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pratelli, Luca; Rigo, Pietro. On the existence of continuous processes with given one-dimensional distributions. Electron. Commun. Probab. 24 (2019), paper no. 46, 9 pp. doi:10.1214/19-ECP255. https://projecteuclid.org/euclid.ecp/1566957630


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References

  • [1] Ambrosio, L., Gigli, N. and Savaré, G.: Gradient flows. Second edition. Birkhauser, Basel, 2008.
  • [2] Blackwell, D. and Dubins, L.E.: An extension of Skorohod’s almost sure representation theorem. Proc. Amer. Math. Soc. 89, (1983), 691–692.
  • [3] Bogachev, V.I. and Kolesnikov, A.V.: Open mappings of probability measures and the Skorohod representation theorem. Theory Probab. Appl. 46, (2002), 20–38.
  • [4] Cortissoz, J.: On the Skorokhod representation theorem. Proc. Amer. Math. Soc. 135, (2007), 3995–4007.
  • [5] Fernique, X.: Un modèle presque sûr pour la convergence en loi. C. R. Acad. Sci. Paris, Sér. 1 306, (1988), 335–338.
  • [6] Letta, G. and Pratelli, L.: Le théorème de Skorohod pour des lois de Radon sur un espace métrisable. Rendiconti Accademia Nazionale delle Scienze detta dei XL 115, (1997), 157–162.
  • [7] Puccetti, G. and Wang, R.: Extremal dependence concepts. Statist. Science 30, (2015), 485–517.
  • [8] Suo, Y., Tao, J. and Zhang, W.: Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth. Front. Math. China 13, (2018), 913–933.