Approximation of Discrete Measures by Finite Point Sets

TL;DR

This paper investigates how well finite point sets can approximate discrete measures on [0,1], establishing that the optimal order of approximation is proportional to 1/N, which confirms the best possible rate for finitely supported discrete measures.

Contribution

It provides a complete characterization of the approximation order for discrete measures, proving the optimality of the 1/N rate for finitely supported cases.

Findings

For discrete measures, the approximation order is bounded below by 1/(cN) for some c ≥ 2.

The known 1/N approximation rate is optimal for finitely supported discrete measures.

The paper fills a gap by extending the understanding of measure approximation to include discrete cases.

Abstract

For a probability measure $\mu$ on $[0,1]$ without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is $\frac{1}{2N}$ as has been proven relatively recently. However, if $\mu$ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on $[0,1]$ by finite point sets has so far not been completely covered in the existing literature. In this note, we close this gap by giving a complete description of the discrete case. Most importantly, we prove that for any discrete measure the best possible order of approximation is for infinitely many $N$ bounded from below by $\frac{1}{cN}$ for some constant $c \geq 2$ which depends on the measure. This implies, that for a finitely supported discrete measure on $[0,1]^d$ the known possible order of approximation $\frac{1}{N}$ is indeed the optimal one.