TY - JOUR
T1 - Asymptotics of One-Dimensional Lévy Approximations
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
UR - http://www.scopus.com/inward/record.url?scp=85064280387&partnerID=8YFLogxK
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 -