On the Expected ℒ₂–Discrepancy of Jittered Sampling

For m, d ∈ 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 ℒ₂−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 ℒ₂−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ₂−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.