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 (λ, t, m, s)-nets and (i, 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.