The asymptotics of purely atomic one-dimensional Levy approximations

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.