The Mean Square Discrepancy of Scrambled (t,s)-Sequences

The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled (,t,m,s)-nets and (t,s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n-1 [log n](s-1)/2) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n-3/2 [log n](s-1)/2) for scrambled nets. For an arbitrary number of points taken from a (t,s)-sequence, the root mean square discrepancy appears to be no better than O(n-1 [log n](s-1)/2), regardless of the smoothness of the reproducing kernel.