<abstract xmlns="http://www.w3.org/1999/xhtml">

For <italic>m, d</italic> ∈ ℕ, a jittered sample of <italic>N</italic> = <italic>m</italic>d points can be constructed by partitioning [0, 1]d into <italic>m</italic>d axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any <italic>m, d</italic> ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.
</abstract>

For m, d ∈ ℕ, a jittered sample of N = md points can be constructed by partitioning [0, 1]d into md axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.

On the Expected ℒ2–Discrepancy of Jittered Sampling

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the Expected ℒ–Discrepancy of Jittered Sampling

For m, d ∈ ℕ, a jittered sample of N = md points can be constructed by partitioning [0, 1]d into md axis-aligned equivolume boxes and placing one point...

On the Expected ℒ₂–Discrepancy of Jittered Sampling

Published Online: Aug 10, 2023

Page range: 65 - 82

Received: Sep 16, 2022

Accepted: Feb 27, 2023

DOI: https://doi.org/10.2478/udt-2023-0005

Keywords
jittered sampling, star-discrepancy, ℒ-discrepancy, Hickernell ℒ-discrepancy

© 2023 Nathan Kirk et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the Expected ℒ2–Discrepancy of Jittered Sampling

Published Online: Aug 10, 2023

Page range: 65 - 82

Received: Sep 16, 2022

Accepted: Feb 27, 2023

DOI: https://doi.org/10.2478/udt-2023-0005

Keywordsjittered sampling, star-discrepancy, ℒ-discrepancy, Hickernell ℒ-discrepancy

© 2023 Nathan Kirk et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the Expected ℒ₂–Discrepancy of Jittered Sampling

Keywords
jittered sampling, star-discrepancy, ℒ-discrepancy, Hickernell ℒ-discrepancy