On the expected L2-discrepancy of jittered sampling

Abstract

For $m,d\in \mathbb{N}$, a jittered sample of $N=m^d$ points can be constructed by partitioning $[0,1{]}^d$ into $m^d$ axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ${\mathcal{L}}_2-$discrepancy of stratified samples stemming from general equivolume partitions of $[0,1{]}^d$ which recently appeared, to derive a closed form expression for the expected ${\mathcal{L}}_2-$discrepancy of a jittered point set for any $m,d\in \mathbb{N}$. As a second main result we derive a similar formula for the expected Hickernell ${\mathcal{L}}_2-$discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.

11 pages, 2 figures. Added the result about Hickernell discrepancy