The dispersion of the Hammersley Sequence in the unit square
Copyright policy )The dispersion of the Hammersley Sequence in the unit square
The notion of dispersion, a measure of denseness of sequences, plays an important role in quasi-Monte Carlo optimization. In this paper, we obtain an explicit formula for the dispersion of an important low dispersion sequence, namely the Hammersley Sequence in the unit square. The dispersiondMof theM points of this sequence, whereM=2N withN a positive integer is given by $$d_M = \frac{{\sqrt {2M - 2\sqrt M + 1} }}{M},if N is even, d_M = \frac{{\sqrt {\left( {5/2} \right)M - \sqrt {8M} + 1} }}{M},if N is odd.$$ .
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- University of the West Indies System Jamaica
- University of the West Indies Jamaica
510.mathematics, explicit formula, Irregularities of distribution, discrepancy, dispersion of Hammersley sequence, Monte Carlo methods, quasi-Monte Carlo optimization, General theory of distribution modulo \(1\), Article
510.mathematics, explicit formula, Irregularities of distribution, discrepancy, dispersion of Hammersley sequence, Monte Carlo methods, quasi-Monte Carlo optimization, General theory of distribution modulo \(1\), Article
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