Several problems in discrepancy theory

: lower bounds and stratified sampling

Abstract

The aim of this PhD thesis is to study several problems in the theory of uniform distribution. Specifically in the subfield of discrepancy theory, which is often referred to in the literature by the theory of irregularities of distribution. We study several problems involving various measures of irregularity of distribution and the subsequent discrepancy values associated with these measures. While discrepancy theory can be discussed in many settings, we contain our study to the classical setting of point sets contained inside the d−dimensional unit hypercube.

As a first contribution, the 1986 results and proofs of Petko Proinov on lower bounds of a particular discrepancy measure named the diaphony are contained in Chapter 1. These methods are written in a self-contained and complete manner for the first time in English. In addition, we discuss the progress since 1986 and provide updated state-of-the-art constants associated with the diaphony.

Chapters 2 to 4 change our focus and are used to extend the recent study of stratified sampling by Markus Kiderlen and Florian Pausinger. On the way, we solve several open problems relating to partitions of the d−dimensional unit cube. In Chapter 2, we derive several closed formulae which give the expected discrepancy values for a specific formulation of stratified sampling in the d−dimensional unit cube called jittered sampling. In particular, we study the expected L2−discrepancy and the expected Hickernell L2−discrepancy of arbitrary jittered sampling in the hypercube.

Chapter 3 is dedicated to the investigation of the expected discrepancy of the stratified point sets obtained from a more general family of partitions. These N−set partitions are constructed via placing N-1 hyperplanes along the main diagonal of the cube in such a way as to create equal volume strata. We provide a recommendation for the construction of this partition of the hypercube after the difficulty of construction in dimension greater than 2 was pointed out by Kiderlen and Pausinger and thereafter, utilise this construction to compute numerical values of the expected discrepancy of the resulting stratified point sets.

Finally, Chapter 4 explores the validity and dependability of using a novel method involving the theory of majorisations to compare the expected discrepancy of stratified point sets obtained from a given pair of partitions. The primary advantage of this method is the fact that one would not be required to explicitly compute the expected discrepancy values to investigate which stratified point set has more regular distribution. We discuss the successes and failures of this method while stating several open problems for this new direction of research.

Date of Award	Jul 2023
Original language	English
Awarding Institution	Queen's University Belfast
Sponsors	UK Research and Innovation
Supervisor	David Barnes (Supervisor), Ying-Fen Lin (Supervisor) & Florian Pausinger (Supervisor)