On the mean square weighted L2 discrepancy of randomized digital nets in prime base

Authors:
Ligia L. Cristea;Josef Dick;Friedrich Pillichshammer
Affiliations:
Institut für Finanzmathematik, Universität Linz, Linz, Austria;School of Mathematics, University of New South Wales, Sydney, Australia;Institut für Finanzmathematik, Universität Linz, Linz, Austria
Venue:
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Year:
2006

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Abstract

We study the mean square weighted L2 discrepancy of randomized digital (t, m, s)-nets over Zp. The randomization method considered here is a digital shift of depth m, i.e., for each coordinate the first m digits of each point are shifted by the same shift, whereas the remaining digits in each coordinate are shifted independently for each point. We also consider a simplified version of this shift.We give a formula for the mean square weighted L2 discrepancy using the generating matrices of the digital net and we prove an upper bound on this discrepancy. Further we investigate how the constant of the leading term depends on the choice of the base p.