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A Skorohod representation theorem for uniform distance
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  • Published: 11 March 2010

A Skorohod representation theorem for uniform distance

  • Patrizia Berti1,
  • Luca Pratelli2 &
  • Pietro Rigo3 

Probability Theory and Related Fields volume 150, pages 321–335 (2011)Cite this article

  • 180 Accesses

  • 6 Citations

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Abstract

Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ~ μ n for all n ≥ 0 and ||X n − X 0|| → 0 in probability, where ||·|| is the sup-norm. Such conditions do not require μ 0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.

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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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Corresponding author

Correspondence to Pietro Rigo.

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Cite this article

Berti, P., Pratelli, L. & Rigo, P. A Skorohod representation theorem for uniform distance. Probab. Theory Relat. Fields 150, 321–335 (2011). https://doi.org/10.1007/s00440-010-0279-6

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  • Received: 27 October 2009

  • Revised: 28 January 2010

  • Published: 11 March 2010

  • Issue Date: June 2011

  • DOI: https://doi.org/10.1007/s00440-010-0279-6

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Keywords

  • Cadlag function
  • Exchangeable empirical process
  • Separable probability measure
  • Skorohod representation theorem
  • Uniform distance
  • Weak convergence of probability measures

Mathematics Subject Classification (2000)

  • 60B10
  • 60A05
  • 60A10
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