A Sharp Discrepancy Bound for Jittered Sampling

Abstract

For $m, d \in {\mathbb N}$, 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 $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c \ge 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies \[c \,dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)}.\] This discrepancy is thus smaller by a factor of $\Theta\big(\sqrt{\frac{1+\log(m/d)}{m/d}}\,\big)$ than 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 $d$.