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Uniformly converging random variables for weakly converging laws
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  • Published: August 2002

Uniformly converging random variables for weakly converging laws

  • Daniel Dubischar1 

Probability Theory and Related Fields volume 123, pages 601–605 (2002)Cite this article

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Abstract.

 Let {P n , n ?ℕ} be a sequence of Borel probability measures on a compact and connected metric space X. We show that in case the measures P n converge weakly to a fully supported limit measure P, there exist uniformly converging random variables X n , n ?ℕ with these given laws. Connectivity and compactness are necessary conditions for our theorem to hold. We also present a decent generalization. We prove our theorem by means of a comparison of the Prokhorov and the so-called minimal L ∞ metric. Then we only need to use the Strassen-Dudley theorem and Kellerer's measure extension theorem for decomposable families.

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Authors and Affiliations

  1. Institut für dynamische Systeme, Universität Bremen, Postfach 330440, 28334 Bremen; and Vorkampsweg 166F, 28359 Bremen, Germany. e-mail: daniel.dubischar@hannover-re.com, , , , , , DE

    Daniel Dubischar

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  1. Daniel Dubischar
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Received: 2 November 2000 / Revised version: 5 January 2002/ Published online: 1 July 2002

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Dubischar, D. Uniformly converging random variables for weakly converging laws. Probab Theory Relat Fields 123, 601–605 (2002). https://doi.org/10.1007/s004400200199

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  • Issue Date: August 2002

  • DOI: https://doi.org/10.1007/s004400200199

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Keywords

  • Probability Measure
  • Limit Measure
  • Extension Theorem
  • Borel Probability Measure
  • Measure Extension
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