A Sharp Discrepancy Bound for Jittered Sampling

Abstract

For m, d ∈ N, a jittered sampling point set P having N = m d points in [0, 1) d is constructed by partitioning the unit cube [0, 1) d into m d axisaligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants c ≥ 0 and C such that for all d and all m ≥ d the expected non-normalized star discrepancy of a jittered sampling point set satisfiesThis discrepancy is thus smaller by a factor of Θthan the one of a uniformly distributed random point set of m d points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity ( 2016)). It also removes the asymptotic requirement that m is sufficiently large compared to d.

On a partition with a lower expected $\mathcal{L}_2$-discrepancy than classical jittered sampling

“…random points; see also [13]. Moreover, Doerr [4] recently determined the precise asymptotic order of the expected star-discrepancy of a point set obtained from jittered sampling:…”

On a partition with a lower expected L2-discrepancy than classical jittered sampling

“…uniform random points in [0, 1] d is of order Θ( 1/N ); see [10] for the first upper bound, [1] for the first upper bound with explicit constant and [4] for the first lower bound as well as [9] for the current state of the art results in this context. This can be compared to a recent result by Doerr [5] on the precise asymptotic order of the expected star-discrepancy of a point set obtained from jittered sampling:…”