Low-discrepancy sequences: theory and applications

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. More precisely, one considers a uniformly distributed sequence (u.d.) (xn)n2N in the unit interval, i.e. a sequence satisfying the relation lim N!1 1 N XN n=1 f(xn) = Z 1 0 f(x)dx ; for every continuous function f de ned on [0; 1]. This relation suggests the possibility of a numerical approximation of the integral on the right-hand side by means of u.d. sequences, even if it does not give any information on the quality of the estimator. The following quantity D N = D N (x1; : : : ; xN) = sup a2[0;1[

PN n=1 1[0;a[(xn) N 􀀀 ([0; a[)

; called the star discrepancy, has been introduced in order to have a quantitative insight on the rate of convergence. One of the most important results about the integration error of this approximation technique is given by the Koksma-Hlawka inequality which states that the error can be bounded by the product of the variation of f (in the sense of Hardy and Krause), denoted by V (f), and the star-discrepancy D N of the point sequence (xn)n2N: 1 N XN n=1 f(xn) 􀀀 Z 1 0 f(x)dx

V (f)D N (xn) : Thus in order to minimize the integration error we have to use point sequences with small discrepancy, that is, sequences which achieve a stardiscrepancy of order O(N􀀀1(logN)). These sequences are called low-discrepancy sequences and they turn out to be very useful especially for the approximation of multidimensional integrals. In this context, the error in the approximation is smaller than the probabilistic one of the standard Monte Carlo method, where a sequence of random points instead of deterministic points, is used. Methods using low-discrepancy sequences, often called quasi-random sequences, are called Quasi-Monte Carlo methods (QMC). However, to construct low-discrepancy sequences, especially multidimensional ones, and to compute the discrepancy of a given sequence are in general not easy tasks. The aim of this thesis is to provide a full description of these problems as well as the methods used to handle them. The idea was to consider tools from ergodic theory in order to produce new low-discrepancy sequences. The starting point to do this, is to look at the orbit, i.e. the sequence of iterates, of a continuous transformation T de ned on the unit interval which has the property of being uniquely ergodic. The unique ergodicity of the transformation has the following consequence: If T : [0; 1] ! [0; 1] is uniquely ergodic, then for every f 2 L1(X) lim N!1 1 N NX􀀀1 n=0 f(Tnx) = Z X f(x)d (x) ; for every x 2 X. So the orbit of x under T is a uniformly distributed sequence. In this respect, we devoted the rst chapter entirely on classical topics in uniform distribution theory 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 a countable family of low-discrepancy sequences, namely the LS-sequences of points. In particular, one of these sequences will be considered in full detail in Chapter 3. The content of Chapters 3, 4 and 5 is based on three published papers that I co-authored. 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 (a weaker property than unique ergodicity), 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 de ned by linear recurrences. In this way we could not only prove that the transformation T de ned in Chapter 3 is uniquely ergodic, but we could also construct multidimensional uniformly distributed sequences. Finally, we go back to the problem of nding an approximation for integrals. In Chapter 5 we tried to nd bounds for integrals of two-dimensional, piecewise constant functions with respect to copulas. Copulas are functions that can be viewed as asymptotic distribution functions with uniform margins. To solve this problem, we had to draw a connection to linear assignment problems, which can be solved e ciently in polynomial time. The approximation technique was applied to problems in nancial mathematics and uniform distribution theory.