Abstract
We consider the b-adic diaphony as a tool to measure the uniform distribution of sequences, as well as to investigate pseudo-random properties of sequences. The study of pseudo-random properties of uniformly distributed nets is extremely important for quasi-Monte Carlo integration. It is known that the error of the quasi-Monte Carlo integration depends on the distribution of the points of the net. On the other hand, the b-adic diaphony gives information about the points distribution of the net.
Several particular constructions of sequences (x i ) are considered. The b-adic diaphony of the two dimensional nets {y i = (x i , x i + 1)} is calculated numerically. The numerical results show that if the two dimensional net {y i } is uniformly distributed and the sequence (x i ) has good pseudo-random properties, then the value of the b-adic diaphony decreases with the increase of the number of the points. The analysis of the results shows a direct relation between pseudo-randomness of the points of the constructed sequences and nets and the b-adic diaphony as well as the discrepancy.
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Lirkov, I., Stoilova, S. (2011). The b-adic Diaphony as a Tool to Study Pseudo-randomness of Nets. In: Dimov, I., Dimova, S., Kolkovska, N. (eds) Numerical Methods and Applications. NMA 2010. Lecture Notes in Computer Science, vol 6046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18466-6_7
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DOI: https://doi.org/10.1007/978-3-642-18466-6_7
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