Families of Well Approximable Measures

Fairchild, Samantha; Goering, Max; Weiß, Christian

doi:10.2478/udt-2021-0003

Cited by 3 publications

(4 citation statements)

References 16 publications

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“…which is the second claim of Theorem 2 (i). This observation also explains entirely, what happens in Example 3.1 of [FGW21], where ξ 1 = ξ 2 = 1 2 .…”

Section: Introductionsupporting

confidence: 63%

“…It is possible to use our approach also in higher dimensions and we again obtain a lower bound of the form D * N (µ, ν N ) ≥ 1 cN for infinitely many N . Together with [FGW21], Proposition 2.2, this leads to the following interesting corollary.…”

Section: Introductionmentioning

confidence: 64%

“…According to [FGW21], the Lebesgue measure is the hardest Borel measure on [0, 1] to approximate by a finite point set. In order to formulate the result in a mathematically precise way, recall first that the star-discrepancy between two probability measures µ, ν on [0, 1] is defined by…”

Section: Introductionmentioning

confidence: 99%

“…In other words, in any dimension any finitely supported discrete measure can be approximated by a finite set of order 1 N and this is the best possible order of approximation. If the decay rate of the weights is fast enough, it is furthermore known due to [FGW21], Theorem 1.1, that certain measures can be approximated by order of convergence at most log(N )/N . Nonetheless, we expect that due to the combinatorial richness of inclusion of half-open intervals in higher dimensions, there exist infinitely supported discrete measures in dimensions d ≥ 3 with a bigger minimal possible order of approximation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Approximation of Discrete Measures by Finite Point Sets

Weiß¹

2022

Preprint

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For a probability measure µ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is 1 2N as has been proven relatively recently. However, if µ 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 1 cN for some constant c ≥ 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 1 N is indeed the optimal one.

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“…which is the second claim of Theorem 2 (i). This observation also explains entirely, what happens in Example 3.1 of [FGW21], where ξ 1 = ξ 2 = 1 2 .…”

Section: Introductionsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximation of Discrete Measures by Finite Point Sets

Weiß¹

2022

Preprint

Self Cite

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On the discrepancy of low-dimensional probability measures

Weiss

2024

Theor. Probability and Math. Statist.

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Calculating the star-discrepancy of a point set in the d d -dimensional unit cube with respect to the Lebesgue measure is an NP-hard problem. Still explicit formulas, which allow for an easy implementation, have been derived. These formulas are particularly compact in the case of dimensions d = 1 , 2 d=1,2 by the work of Niederreiter, and Bundschuh and Zhu. In this paper, we generalize their formulas to arbitrary measures in the dimension d = 1 d=1 and to a wide class of measures in the dimension d = 2 d=2 . In order to give a potential application of such formulas, we reprove from it the fact that the Lebesgue measure is the hardest measure to approximate in the dimension d = 1 d=1 .

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Approximation of Discrete Measures by Finite Point Sets

Weiß

2023

Uniform Distribution Theory

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For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &inline as has been proven relatively recently. However, if μ 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 the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by &inline for some constant 6 ≥ c> 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 &inline is indeed the optimal one.

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Families of Well Approximable Measures

Cited by 3 publications

References 16 publications

Approximation of Discrete Measures by Finite Point Sets

Approximation of Discrete Measures by Finite Point Sets

On the discrepancy of low-dimensional probability measures

Approximation of Discrete Measures by Finite Point Sets

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