The asymptotic efficiency of randomized nets for quadrature

An L-2-type discrepancy arises in the average- and worst-case error analyse s for multidimensional quadrature rules. This discrepancy is uniquely defin ed by K(x, y), which serves as the covariance kernel for the space of rando m functions in the average-case analysis and a reproducing kernel for the s pace of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0, m, s)-nets in base b. For moderately smooth K(x, y) the discrepancy is O(N-1 [log(N)]((s-1)/2)), and for EC(x, y) with greater smoothness the discrepanc y is O(N-3/2 [log(N)]((s-1)/2)), where N = b(m) is the number of points in the net. Numerical experiments indicate that the (t, m, s)-nets of Faure, N iederreiter and Sober do not necessarily attain the higher order of decay f or sufficiently smooth kernels. However, Niederreiter nets may attain the h igher order for kernels corresponding to spaces of periodic functions.