AbstractWe extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$
Ω
=
(
Ω
1
,
…
,
Ω
N
)
be a partition of $$[0,1]^d$$
[
0
,
1
]
d
and let the ith point in $${{\mathcal {P}}}$$
P
be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$
i
=
1
,
…
,
N
. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$
L
p
-discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$
E
L
p
(
P
Ω
)
p
, of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$
P
Ω
generated from any equivolume partition $${\varvec{\Omega }}$$
Ω
is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$
L
p
-discrepancy of a set of N uniform random samples for $$p>1$$
p
>
1
. For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$
L
p
-discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.
Facial structure of strongly convex sets generated by random samples
