High-discrepancy sequences

First, it is pointed out that the uniform distribution of points in [0, 1]^d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]^d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]^d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N^-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.