Variance bounds and existence results for randomly shifted lattice rules

Abstract

We study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions. We consider integrands that are square integrable but whose Fourier series are not necessarily absolutely convergent. For such integrands, a bound on the variance is expressed through a certain type of weighted discrepancy. We prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost O(n−α), where n is the number of function evaluations and α>1 depends on our assumptions on the convergence speed of the Fourier coefficients. These results hold for general weights, arbitrary n, and any dimension. With additional conditions on the weights, we obtain a convergence that holds uniformly in the dimension, and this provides sufficient conditions for strong tractability of the integration problem. We also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms.