Asymptotics of One-Dimensional Lévy Approximations

Arno Berger; Chuang Xu

doi:10.1007/s10959-019-00893-1

Asymptotics of One-Dimensional Lévy Approximations

Arno Berger^*, Chuang Xu

^*Corresponding author for this work

Department of Mathematical Sciences

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.

Original language	English
Journal	Journal of Theoretical Probability
ISSN	0894-9840
DOIs	https://doi.org/10.1007/s10959-019-00893-1
Publication status	Published - 1 Jun 2020

Keywords

Approximation error
Asymptotic point distribution
Best (uniform) approximation
Inverse function
Inverse measure
Lévy probability metric

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10.1007/s10959-019-00893-1

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title = "Asymptotics of One-Dimensional L{\'e}vy Approximations",

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 L{\'e}vy 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 L{\'e}vy approximations are more complete and require fewer assumptions.",

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doi = "10.1007/s10959-019-00893-1",

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AU - Berger, Arno

AU - Xu, Chuang

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

KW - Approximation error

KW - Asymptotic point distribution

KW - Best (uniform) approximation

KW - Inverse function

KW - Inverse measure

KW - Lévy probability metric

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U2 - 10.1007/s10959-019-00893-1

DO - 10.1007/s10959-019-00893-1

M3 - Journal article

AN - SCOPUS:85064280387

SN - 0894-9840

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

ER -

Asymptotics of One-Dimensional Lévy Approximations

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Keywords

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