On the root mean square weighted \(L_{2}\) discrepancy of scrambled nets (Q1888373)

The author proves an estimate for the root mean square weighted \(L_{2}\) discrepancy of scrambled \((0,m,s)\)-nets in base \(b\) which are frequently used quasi-Monte Carlo (QMC) point sets. For a net of size \(N\), the bound is essentially (i.e. up to logarithmic factors) proportional to \(N^{-1}\) which is known to be the optimal order. In contrast to this, Monte-Carlo points exhibit a weighted \(L_{2}\) discrepancy proportional to \(N^{-1/2}\). For certain sets of weights, the bound is explicitly computed. Unfortunately the dependence on the dimension is exponentially. Nevertheless, the bound can serve as substitute for explicit expressions for the \(L_{2}\) discrepancy which are only computable for fairly low dimensions.