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.
Author information
Authors and Affiliations
Additional information
Received: 2 November 2000 / Revised version: 5 January 2002/ Published online: 1 July 2002
Rights and permissions
About this article
Cite this article
Dubischar, D. Uniformly converging random variables for weakly converging laws. Probab Theory Relat Fields 123, 601–605 (2002). https://doi.org/10.1007/s004400200199
Issue Date:
DOI: https://doi.org/10.1007/s004400200199
Keywords
- Probability Measure
- Limit Measure
- Extension Theorem
- Borel Probability Measure
- Measure Extension