Asymptotics of One-Dimensional Lévy Approximations

Arno Berger () and Chuang Xu
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Arno Berger: University of Alberta
Chuang Xu: University of Copenhagen

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 1164-1195

Abstract: Abstract For arbitrary Borel probability measures on the real line, necessary and sufficient conditions are presented that characterize best purely atomic approximations relative to the classical Lévy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms. The precise asymptotics (as the number of atoms goes to infinity) of the approximation error is identified for the important special cases of best uniform (i.e. all atoms having equal weight) and best (i.e. unconstrained) approximations, respectively. When compared to similar results known for other probability metrics, the results for Lévy approximations are more complete and require fewer assumptions.

Keywords: Best (uniform) approximation; Lévy probability metric; Inverse function; Inverse measure; Approximation error; Asymptotic point distribution; 60B10; 60E15; 62E15 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00893-1

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