An average discrepancy for optimal vertex-modified number-theoretic rules

Number-theoretic rules are particularly suited for the approximation of multi-dimensional integrals in which the integrands are periodic. When the integrands are not periodic, then a vertex-modiied variant has been proposed. An error bound for such vertex-modiied rules is based on a simple generalization of the L2 discrepancy. In s dimensions these vertex-modiied rules contain 2 s weights which may be chosen optimally so that the discrepancy is minimized. We obtain an expression for an average value of the squared L2 discrepancy for these optimal vertex-modiied number-theoretic rules. Values of this average may be compared with the corresponding average for normal number-theoretic rules and the expected value for Monte Carlo rules.