The asymptotic efficiency of randomized nets for quadrature

An $\mathcal{L}_{2}$-type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by $K(x,y)$, which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space 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 $\operatorname{O}(N^{-1}[\log(N)]^{(s-1)/2})$, and for $K(x,y)$ with greater smoothness the discrepancy is $\operatorname{O}(N^{-3/2}[\operatorname{log}(N)]^{(s-1)/2})$, where $N=b^{m}$ is the number of points in the net. Numerical experiments indicate that the $(t,m,s)$-nets of Faure, Niederreiter and Sobol' do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.