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The asymptotics of purely atomic one-dimensional Levy approximations
Probability · Submitted 20 Sep 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 Levy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms.
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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 Levy 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 (unconstrained) approximations, respectively. When compared to similar results known for other probability metrics, the results for Levy approximations are more complete and require fewer assumptions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 Levy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms. | contribution 0.57 | 0.91 | 0.64 | asserted 1.00 |
| s1 | 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 (unconstrained) approximations, respectively. | result 0.97 | 0.93 | 0.33 | asserted 1.00 |
| s2 | When compared to similar results known for other probability metrics, the results for Levy approximations are more complete and require fewer assumptions. | interpretation 0.89 | 0.51 | 0.14 | asserted 1.00 |
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Original 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 Levy 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 (unconstrained) approximations, respectively. When compared to similar results known for other probability metrics, the results for Levy approximations are more complete and require fewer assumptions.
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The result is reminiscent of a theorem regarding best uniform $d_W$-approximations [ref] (see also [ref] ), but unlike in that theorem, no integrability assumption on $d μ^-1/dλ$ is needed, and the limit in question always is finite.
Best finite constrained approximations of one-dimensional probabilities · Best uniform Lévy approximations · 1.00
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The result is reminiscent of a theorem regarding best uniform $d_W$-approximations [ref] (see also [ref] ), but unlike in that theorem, no integrability assumption on $d μ^-1/dλ$ is needed, and the limit in question always is finite.
Uniform decomposition of probability measures: quantization, classification, rate of convergence · Best uniform Lévy approximations · 1.00
Cites as background 1 paper
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Also, $μ^-1 := λ_F_μ^-1$ is a positive Borel measure supported on $G_F_μ^-1⊂[0,1]$, referred to as the inverse measure of $μ$; see, e.g., [ref] .
Best finite constrained approximations of one-dimensional probabilities · Best uniform Lévy approximations · 0.96
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