On the mean square weighted [formula omitted] discrepancy of randomized digital nets in prime base

Abstract

We study the mean square weighted L 2 discrepancy of randomized digital ( t , m , s ) -nets over Z p . 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 L 2 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.