Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands

Abstract

We study the multivariate integration problem ∫ R d f ( x ) ρ ( x ) d x , with ρ being a product of univariate probability density functions. We assume that f belongs to a weighted tensor-product reproducing kernel Hilbert space of functions whose mixed first derivatives, when multiplied by a weight function ψ , have bounded L 2 -norms. After mapping into the unit cube [ 0 , 1 ] d , the transformed integrands are typically unbounded or have huge derivatives near the boundary, and thus fail to lie in the usual function space setting where many good results have been established. In our previous work, we have shown that randomly shifted lattice rules can be constructed component-by-component to achieve a worst case error bound of order O ( n − 1 / 2 ) in this new function space setting. Using a more clever proof technique together with more restrictive assumptions, in this article we improve the results by proving that a rate of convergence close to the optimal order O ( n − 1 ) can be achieved with an appropriate choice of parameters for the function space. The implied constants in the big- O bounds can be independent of d under appropriate conditions on the weights of the function space.