We study the discrepancy of jittered sampling sets: such a set Pβ[0,1]d is generated for fixed mβN by partitioning
[0,1]d into md axis aligned cubes of equal measure and placing a random
point inside each of the N=md cubes. We prove that, for N sufficiently
large, 101βN21β+2d1βdββ€EDNββ(P)β€N21β+2d1βdβ(logN)21ββ, where the upper bound with an unspecified constant Cdβ
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 Nβ³dd. 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 L2βdiscrepancy is smaller than that of
purely random points