On the discrepancy of jittered sampling

We study the discrepancy of jittered sampling sets: such a set P subset of [0, 1](d) is generated for fixed m is an element of N by partitioning [0, 1](d) into and axis aligned cubes of equal measure and placing a random point inside each of the N = m(d) cubes. We prove that, for N sufficiently large, 1/10 d/N 1/2 + 1/2d <= EDN*(P) <= root d(log N)1/2/N 1/2 + 1/2d, where the upper bound with an unspecified constant C-d was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N greater than or similar to d(d). We also prove a partition principle showing that every partition of [0, 1](d) combined with a jittered sampling construction gives rise to a set whose expected squared L-2-discrepancy is smaller than that of purely random points. (C) 2015 Elsevier Inc. All rights reserved.