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Skorohod representation on a given probability space
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  • Published: 26 July 2006

Skorohod representation on a given probability space

  • Patrizia Berti1,
  • Luca Pratelli2 &
  • Pietro Rigo3 

Probability Theory and Related Fields volume 137, pages 277–288 (2007)Cite this article

  • 210 Accesses

  • 10 Citations

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Abstract

Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n  = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).

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References

  1. Berti P., Pratelli L., Rigo P.(2004). Limit theorems for a class of identically distributed random variables. Ann. Probab. 32, 2029–2052

    Article  MathSciNet  Google Scholar 

  2. Berti P., Pratelli L., Rigo P.(2006). Asymptotic behaviour of the empirical process for exchangeable data. Stoch. Proc. Appl. 116, 337–344

    Article  MathSciNet  Google Scholar 

  3. Crimaldi, I., Letta, G., Pratelli, L.: A strong form of stable convergence. Sem Probab (to appear) (2005)

  4. Dudley R.(1999). Uniform central limit theorems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Hall P., Heyde C.C.(1980). Martingale limit theory and its applications. Academic, New York

    Google Scholar 

  6. Letta, G., Pratelli, L.: Le théorème de Skorohod pour des lois de Radon sur un espace métrisable. In: Rendiconti Accademia Nazionale delle Scienze detta dei XL, vol. 115, pp. 157–162 (1997)

  7. Renyi A.(1963). On stable sequences of events. Sankhya A 25, 293–302

    MathSciNet  Google Scholar 

  8. van der Vaart A., Wellner J.A.(1996). Weak convergence and empirical processes. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  9. van der Vaart A., van Zanten H.(2005). Donsker theorems for diffusions: necessary and sufficient conditions. Ann. Probab. 33, 1422–1451

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata “G. Vitali”,, Universita’ di Modena e Reggio-Emilia, via Campi 213/B, 41100, Modena, Italy

    Patrizia Berti

  2. Accademia Navale, viale Italia 72, 57100, Livorno, Italy

    Luca Pratelli

  3. Dipartimento di Economia Politica e Metodi Quantitativi, Universita’ di Pavia, via S. Felice 5, 27100, Pavia, Italy

    Pietro Rigo

Authors
  1. Patrizia Berti
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  2. Luca Pratelli
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  3. Pietro Rigo
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Correspondence to Pietro Rigo.

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Berti, P., Pratelli, L. & Rigo, P. Skorohod representation on a given probability space. Probab. Theory Relat. Fields 137, 277–288 (2007). https://doi.org/10.1007/s00440-006-0018-1

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  • Received: 06 February 2006

  • Revised: 27 May 2006

  • Published: 26 July 2006

  • Issue Date: March 2007

  • DOI: https://doi.org/10.1007/s00440-006-0018-1

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Keywords

  • Empirical process
  • Non measurable random element
  • Skorohod representation theorem
  • Stable convergence
  • Weak convergence of probability measures

Mathematics Subject Classification (2000)

  • 60B10
  • 60A05
  • 60A10
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