Low discrepancy sequences: Theory and Applications

TL;DR

This thesis explores the asymptotic behavior of sequences modulo 1 using ergodic theory, providing new methods to construct multidimensional low-discrepancy sequences with applications in numerical integration and finance.

Contribution

It introduces novel ergodic and dynamical methods for constructing and analyzing low-discrepancy sequences, advancing the understanding of their uniform distribution properties.

Findings

Proved ergodicity of a transformation generating LS-sequences.

Constructed multidimensional uniformly distributed sequences.

Connected integral bounds with linear assignment problems.

Abstract

The main topic of this present thesis is the study of the asymptotic behaviour of sequences modulo 1. In particular, by using ergodic and dynamical methods, a new insight to problems concerning the asymptotic behaviour of multidimensional sequences can be given, and a criterion to construct new multidimensional uniformly distributed sequences is provided. The starting point to do this, is to look at the orbit, i.e. the sequence of iterates, of a uniquely ergodic transformation T defined on [0,1]. The unique ergodicity of the transformation has the following consequence: the orbit of x under T is a uniformly distributed sequence. We devoted the first chapter entirely on classical topics in uniform distribution theory (UDT) and ergodic theory. This provides the basic requirements for a complete understanding of the following chapters, even to a reader who is not familiar with the subject. Chapter 2 deals with the LS-sequences of points. In Chapter 3, the method used to construct the transformation T is the so-called cutting-stacking technique. In particular, we were able to prove the ergodicity of T and that the orbit of the origin under this map coincides with an LS-sequence which turns out to be a low-discrepancy one. In Chapter 4, another approach from ergodic theory is used. This approach is based on the study of dynamical systems arising from numeration systems defined by linear recurrences. In this way we could not only prove that the transformation T defined in Chapter 3 is uniquely ergodic, but we could also construct multidimensional uniformly distributed sequences. In Chapter 5 we found bounds for integrals of two-dimensional, piecewise constant functions with respect to copulas. To solve this problem, we drew a connection to linear assignment problems. The approximation technique was applied to problems in financial mathematics and UDT.