Uniformly converging random variables for weakly converging laws |
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Authors: | Daniel Dubischar |
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Institution: | Institut für dynamische Systeme, Universit?t Bremen, Postfach 330440, 28334 Bremen; and Vorkampsweg 166F, 28359 Bremen, Germany. e-mail: daniel.dubischar@hannover-re.com, DE
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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.
Received: 2 November 2000 / Revised version: 5 January 2002/ Published online: 1 July 2002 |
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Keywords: | |
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