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.

中文翻译：

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一维 Lévy 近似的渐近性

对于实线上的任意 Borel 概率度量，给出了相对于经典 Lévy 概率度量的最佳纯原子近似的充分必要条件，给定任意数量的原子，并允许对原子的位置或权重进行额外的约束。近似误差的精确渐近线（随着原子数趋于无穷大）分别针对最佳均匀（即所有原子具有相同的权重）和最佳（即无约束）近似的重要特殊情况确定。与其他概率度量已知的类似结果相比，Lévy 近似的结果更完整，需要的假设更少。