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A Skorohod representation theorem for uniform distance

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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Correspondence to Pietro Rigo.

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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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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