On computing the exact value of dispersion of a sequence

Abstract

AbstractWe introduce a method for computing the exact value of the dispersion of a sequence. Dispersion is a measure for the irregularity of distribution of a sequence and plays an important role in the computation of the extreme values of a function by quasi Monte Carlo methods. We use the important two-dimensional Hammersley sequences to illustrate the applicability of the method. We obtain an explicit formula for the dispersion of the Hammersley sequence with arbitrary radix R, where R is an integer greater than or equal to 2. This is a generalization of the result obtained by Peart (1982) for radix 2. With M = RN, N a positive integer, the dispersion dM is given by MdM=2M−2M+1if N is even,2q2(M/R)−2qM/R+1,if N is odd and R is odd,(2q2+2q+1)(M/R)−2(q+1)M/R+1,if N is odd and R is even, where q = [12(R+1)] is the largest integer less than or equal to 12(R+1)