On the Existence of Balancing Allocations and Factor Point Processes

Abstract

In this article, we show that every stationary random measure on R^d that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor. As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures Φ and Ψ with equal intensities. In particular, we prove that such an allocation exists if Φ is diffuse and either (Φ, Ψ) is essentially free or Φ assigns zero measure to every (d − 1)-dimensional affine hyperplane. The main result is deduced from an existing result in descriptive set theory, that is, the existence of lacunary sections. We also weaken the assumption of being essentially free to the case where a discrete group of symmetries is allowed.

Full text

On the Existence of Balancing Allocations and
Factor Point Processes
Ali Khezeli ∗†
, Samuel Mellick ‡
March 9, 2023
Abstract
In this article, we show that every stationary random measure on Rd
that is essentially free (i.e., has no symmetries a.s.) admits a point process
as a factor. As a result, we improve the results of Last and Thorisson
(2022) on the existence of a factor balancing allocation between ergodic
pairs of stationary random measures Φand Ψwith equal intensities. In
particular, we prove that such an allocation exists if Φis diuse and
either (Φ,Ψ) is essentially free or Φassigns zero measure to every (d−
1)-dimensional ane hyperplane. The main result is deduced from an
existing result in descriptive set theory, that is, the existence of lacunary
sections. We also weaken the assumption of being essentially free to the
case where a discrete group of symmetries is allowed.
1 Introduction
Abalancing allocation between measures φand ψon Rdis a map T:Rd→Rd
such that T∗φ=ψ. More generally, a balancing transport is a Markovian kernel
on Rdthat transports φto ψ. For stationary random measures Φand Ψ, one
is interested in the existence of balancing allocations and transports that are
translation-invariant factors of (Φ,Ψ). This goes back to the shift-coupling
theorem of Thorisson [13], extra head schemes for the Poisson point process [4]
and fair tessellations for stationary point processes [5]. In general, it is proved
that invariant balancing transports exist if and only if Φand Ψhave equal
sample intensities [11]. If (Φ,Ψ) is ergodic, this boils down to the equality of
the intensities of Φand Ψ. An explicit construction of an invariant transport is
provided in [3], which is a generalization of [4].
The existence of invariant allocations is more complicated. An invariant
allocation between some multiple of the Lebesgue measure and a point process
∗Inria Paris, ali.khezeli@inria.fr
†Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares
University, P.O. Box 14115-134, Tehran, Iran, khezeli@modares.ac.ir
‡McGill University, samuel.mellick@mcgill.ca
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Ψis equivalent to an invariant fair tessellation; i.e., assigning cells of equal
volume to the points of Ψinvariantly, such that a tessellation of the space is
obtained a.s. Indeed, a fair tessellation always exists for every stationary point
process (see [5]). This is used to construct an extra head scheme for the Poisson
point process; i.e., a point x∈Ψas a function of Ψsuch that (Ψ \ {x})−x
has the same distribution as Ψ(see [5] and [4]). More generally, if Ψis an
arbitrary ergodic random measure, an invariant allocation between a multiple
of the Lebesgue measure and Ψresults in a non-randomized shift-coupling of Ψ
and its Palm version; i.e., a point x∈Rdas a function of Ψsuch that Ψ−x
has the same distribution as the Palm version of Ψ(see [5]).
For the existence of invariant allocations in the general case, it is conve-
nient to assume that Φis diuse; i.e., has no atoms (otherwise, combinatorial
complexities appear). In [12], it is proved that if Φis diuse and there exists
an auxiliary nonempty point process Pas a factor of Φand Ψ(e.g., when Ψ
has atoms), then an invariant balancing allocation exists (under the necessary
conditions mentioned above). This had also been proved in [10] under the extra
assumption that Φassigns zero measure to every (d−1)-dimensional Lipschitz
manifold. This gives rise naturally to the question of the existence of factor
point processes, which is asked in [10] (see also [12]). In this paper, we answer
the problem armatively by proving the following theorem:
Theorem 1.1. Let Φand Ψbe arbitrary random measures on Rd. There exists
a point process as a translation-invariant factor of (Φ,Ψ) (resp. of Φ) that is
nonempty a.s. if and only if (Φ,Ψ) (resp. Φ) has no invariant direction a.s.
Here, an invariant direction of (Φ,Ψ) is a vector t∈Rd\ {0}such that
Φ + λt = Φ and Ψ + λt = Ψ for all λ∈R. Note that if (Φ,Ψ) is essentially free;
i.e., has no translation-symmetry a.s., then there is no invariant direction. Note
also that the set of all invariant directions (plus the origin) is a vector space,
which is called the subspace of invariant directions of (Φ,Ψ).
Using the above theorem, we will prove the following result on the existence
of factor allocations.
Theorem 1.2. Let Φand Ψbe stationary random measures on Rdwith equal
sample intensities. Assume that Φis diuse a.s. If at least one of the following
conditions holds, then there exists an invariant balancing allocation between Φ
and Ψthat is a factor of (Φ,Ψ):
(i) (Φ,Ψ) is essentially free.
(ii) (Φ,Ψ) has no invariant direction.
(iii) Φassigns zero measure to every (d−1)-dimensional ane subspace.
(iv) Φassigns zero measure to every translate of the space of invariant direc-
tions of (Φ,Ψ).
Theorem 1.1 is deduced immediately from a theorem in descriptive set the-
ory, that is, the existence of lacunary sections [8]. This will be discussed in
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Section 3. In order to be self-contained and to present the result to proba-
bilists, a direct proof of Theorem 1.1 will also be given in Section 4. This proof
is a simplication of the existence of lacunary sections in the special case of
random measures on Rd. Also, this theorem can be generalized to the more
general setting of actions of groups. This is closely related to the result of [1]
showing that every essentially free probability-measure-preserving action of a
non-discrete locally compact second-countable group Gis isomorphic to a point
process of nite intensity on G.
Remark 1.3. A few days before publishing this preprint, an independent
preprint [7] is published which proves the existence of factor balancing allo-
cations under stronger conditions. This work assumes that Φassigns zero mea-
sure to every (d−1)-dimensional Lipschitz manifold, which is stronger than
the assumptions of Theorem 1.2 (it is claim in [7] that the condition is sharp,
but Theorem 1.2 shows that this is not the case). The method of the proof is
by using optimal transport and an extension of Monge’s theorem to stationary
random measures provided in [6]. So this work does not prove Theorem 1.1 on
the existence of factor point processes.
2 Denitions
Let Mbe the set of locally-nite Borel measures on Rd. This space is equipped
with the Prokhorov metric dPand is a Polish space. A measure φ∈ M is
diuse if it has no atoms; i.e., φ({x})=0,∀x∈Rd. For t∈Rdand φ∈M,
dene φ+tby (φ+t)(A) := φ(A−t)for A⊆Rd. A stationary random
measure is a random element Φof Msuch that its distribution is invariant
under translations; i.e., Φ + thas the same distribution as Φfor all t∈Rd.
Similarly, a pair of random measures (Φ,Ψ) is called (jointly-) stationary if the
joint distribution is invariant under translations.
Given a measure φ∈ M, the group of translation-symmetries of φis H:=
H(φ) := {t∈Rd:φ−t=φ}. Since His a closed subgroup of Rd, it can
be decomposed into the sum of a linear subspace V:= V(φ)⊆Rdand a
lattice in the orthogonal complement of V. The subspace Vis the subspace
of invariant directions of φ. Similar denitions can be provided for a pair
(φ, ψ)of measures. A stationary random measure is called essentially free if
H(Φ) = {0}a.s. It has no invariant direction a.s. if V(Φ) = 0 a.s.
Let Ibe the σ-eld of invariant events in M; i.e., those event that are
invariant under all translations. Φis called ergodic if every event in Ihas
probability zero or one. Ergodicity of (Φ,Ψ) is dened similarly. The intensity
of Φis dened by E[Φ(C)], where C⊆Rdis an arbitrary Borel set with unit
volume. The sample intensity of Φis the random variable E[Φ(C)|I].
A point process Pis called a (invariant) factor of Φif Pis equal to a
measurable and translation-equivariant function of Φ; i.e., P(Φ + t) = P(Φ) + t
for every t∈Rdand for every sample of Φ. An allocation τ:Rd→Rdis a factor
of Φif it is translation-equivariant function of Φand the map (x, Φ) 7→ τ(x)
(dened on Rd× M) is measurable.
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3 Proof of Theorem 1.1 Using Lacunary Sections
The following denition and result are borrowed from [9]. Let Xbe a Polish
space equipped with a Borel action of a topological group G. A Borel subset
S⊆Xis called a complete section if it intersects every orbit of the action. It
is called a lacunary section if there is a neighborhood Uof the identity of G
such that for every s∈S, one has (U·s)∩S={s}. The following is a special
case of Theorem 3.10 of [9], which is given originally in [8]:
Theorem 3.1 ([8]).Under the above assumptions, if Gis a locally-compact
Polish group, then there exists a complete lacunary section.
We will deduce Theorem 1.1 from the above theorem.
Proof of Theorem 1.1. For simplicity of notations, we prove Theorem 1.1 for
factors of Φonly. The proof for factors of (Φ,Ψ) is identical.
Let Hbe the group of translation-symmetries of Φand Vbe the subspace
of invariant directions. Observe that if V6={0}with positive probability, then
there exists no nonempty factor point process (otherwise, the point process
should also have invariant directions, which is impossible). This proves the if
side of the claim.
For the other side, we will use Theorem 3.1. Consider the action of G:= Rd
on Mby translations. By Theorem 3.1 above, there exists a complete lacunary
section S⊆ M∗for this action. Dene P:= P(Φ) := {t∈Rd: Φ −t∈S}.
Since Sis a complete section, Pis nonempty. In general, Pneed not be a point
process since it contains translated copies of H. However, if His a discrete, it is
straightforward to show that Pis also a discrete set (since Sis lacunary). This
is equivalent to the condition that Vis trivial; i.e., Φhas no invariant direction.
Note that in this case, Pis a translation-invariant factor point process of Φ. So
the claim is proved.
4 Direct Proof of Theorem 1.1
In this section, we provide a direct proof of Theorem 1.1 in order to be self-
contained and to show the idea more clearly. In the case of essential freeness,
this proof is a simplication of the proof of [2] in the special case of random
measures. The proof is slightly generalized to cover the case where Φis not
essentially free.
As in Section 3, we prove Theorem 1.1 for factors of Φonly. The necessity
of the condition is also trivial and is shown in Section 3. So we prove suciency
here.
Direct Proof of Theorem 1.1. Let us start by an outline the strategy of the
proof. We rst produce a factor of Φthat is an invariant random open set
of Rdsuch that it is non-empty with positive probability and its connected
components are bounded. It is then simple to rene this factor open set to a
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point process. Then, a sort of measurable Zorn’s lemma argument shows that
Φmust admit point process factors which are almost surely nonempty.
Let Hbe the group of translation-symmetries of Φand Vbe the subspace
of invariant directions. Assume V={0}a.s. This implies that His a discrete
lattice in Rd. Hence, there exists a smallest natural number N=N(Φ) such
that B2/N (0) ∩H={0}, where Br(0) is the closed ball of radius rcentered at
0. Consider the shell K:= K(Φ) := {t∈Rd:1
N≤ |t| ≤ 2
N}. Given a measure
φ∈ M, dene θKφ:= {φ+t:t∈K(φ)} ⊆ M. The denition of Ngives that
Φ6∈ θKΦa.s. Since θKΦis a closed subset of M, one obtains dP(Φ, θKΦ) >0
a.s. Hence, by choosing ϵ > 0suciently small, one can assume P[Φ ∈Aϵ]>0,
where
Aϵ:= {φ∈ M :dP(φ, θKφ)> ϵ}.
Observe that Aϵis an open subset of M. We may therefore x some open subset
B⊆Aϵof diameter less than ϵsuch that P[Φ ∈B]>0(simply express Aϵas a
countable union of balls of radius less than ϵ, and then pick one that contains
Φwith positive probability). Dene
U:= U(Φ) := {t∈Rd: Φ −t∈B}.
Observe that Uis an open subset of Rdwhich is an equivariant factor of Φ;
i.e., U(φ+s) = U(φ) + s, ∀φ∈ M,∀s∈Rd. It is also nonempty with positive
probability, for instance, 0∈Uif Φ∈B.
The key property of Uis the following: If t, s ∈U, then Φ−tand Φ−s
belong to B, and hence, dP(Φ −t, Φ−s)< ϵ (since Bhas diameter less than
ϵ). Therefore, t−s6∈ Kby the denition of Aϵ. That is, either |t−s|<1
N
or |t−s|>2
N. As a result, the following denes an equivalence relation on U:
t∼sif |t−s|<1
N. Also, every equivalence class has diameter at most 1
N, and
hence, is bounded.
Now, one may produce a factor point process Pin many ways1. For instance,
for each of the (countably many) equivalence classes Cof U, choose the least
element of the closure Caccording to the lexicographic order. Note that this
process is an invariant factor of Φand is uniformly separated.
Observe that Pis nonempty if and only if Φ∈θRdB. Note that θRdBis
invariant, open and has positive probability. If Φwere ergodic, then this would
be an almost sure event and we would be done. In the general case, let p≤1
denote the supremum of P[Φ ∈E], where E⊆ M ranges over all open and Rd-
invariant sets such that there exists a factor point process which is nonempty
on E. It can be seen that the supremum is attained (if P[Φ ∈En]→p, let
Pnbe a factor point process which is nonempty on Enand dene P:= Pnif
Φ∈En\(E1∪. . .∪En−1)). Also, if Eis any such event such that P[Φ ∈E]<1,
then one can enlarge Ea little bit (it is enough to repeat the proof by replacing
Mwith M \ E). These two facts imply that the supremum is attained and
p= 1, which implies the claim of the theorem.
An alternative constructive proof of the last step is by covering Aϵby open
sets B1, B2, . . . ⊆Aϵsuch that each Bihas diameter less than ϵ. Then, modify
1This step is specialized to Rdand its proof is more involved for the actions of other groups.
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the proof by choosing the smallest isuch that Φ∈θRdBi. Also, choose ϵas
an invariant function of Φ; e.g., the largest number of the form ϵ= 1/M such
that Φ∈θRdAϵand P[Aϵ]>0. This way, the constructed point process Pis
nonempty a.s. and the theorem is proved.
5 Proof of Theorem 1.2
In this section, we prove Theorem 1.2 using Theorem 1.1.
Proof of Theorem 1.2. (i). If (Φ,Ψ) is essentially free, then it has no invariant
direction a.s. So the claim is implied by part (ii) proved below.
(ii). If there is no invariant direction a.s., then Theorem 1.1 implies that
there exists a point process that is nonempty a.s. and is a translation-invariant
factor of (Φ,Ψ). Using this as an auxiliary point process, Theorem 1.1 of [12]
constructs a balancing factor allocation.
(iii). The claim is implied by part (iv), which will be proved below. The
only remaining case is when the subspace of invariant directions is the whole
Rd. In this case, both Φand Ψare multiples of the Lebesgue measure and the
claim is trivial.
(iv). Let Vbe the subspace of invariant directions. First, assume that
(Φ,Ψ) is ergodic2. In this case, Vis almost surely equal to a deterministic
subspace (since it is a translation-invariant function of (Φ,Ψ)). Let Wbe the
orthogonal complement of V. The measures Φand Ψinduce measures on W.
More precisely, given an arbitrary Borel set Cin Vwith unit volume, for A⊆W,
dene Φ′(A) := Φ(A⊕C)and Ψ′(A) := Ψ(A⊕C), where A⊕C:= {x+y:
x∈A, y ∈C}is the Minkowski sum of Aand C. Now, Φ′and Ψ′are ergodic
stationary random measures on Wand their intensities are equal to those of
Φand Ψ(and hence, are equal). Also, (Φ′,Ψ′)has no invariant direction. In
addition, the assumption of (iv) implies that Φ′is diuse. Therefore, part (ii)
implies that there exists a balancing allocation between Φ′and Ψ′which is a
factor of (Φ′,Ψ′). If τ′denotes this allocation (note that τ′:W→W), dene
the allocation τon Rdby τ(v+w) := v+τ′(w),∀v∈V, ∀w∈W. Then, τis a
factor allocation which balances between Φand Ψas desired.
In the general case where (Φ,Ψ) might be non-ergodic, the spaces Vand
Wmight be random and some care is needed to choose the allocation as a
measurable factor of (Φ,Ψ) (a naive ergodic decomposition is not sucient).
In this case, construct V, W, Φ′and Ψ′as above. Then, let k:= dim(W)and
choose an orthonormal basis (e1, . . . , ek)for Was a measurable function of W
(e.g., consider the orthogonal projection of the standard unit vectors of Rdon W
and use the Gram–Schmidt algorithm). This denes a linear map L:W→Rk.
Let Φ′′ := L∗Φ′and Ψ′′ := L∗Ψ′. By part (ii), construct a factor balancing
allocation τ′′ between Φ′′ and Ψ′′. Then, dene the allocation τ′between Φ′
and Ψ′by τ′:= L−1◦τ′′ ◦L, and nally, construct τsimilarly to the previous
2In the ergodic case, (iv) is deduced from part (ii) in [10]. For being self-contained, we
include the proof here.
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paragraph. In this construction, every constructed item is a Borel measurable
function of the previously constructed items. This implies that τis a measurable
factor of (Φ,Ψ) and the claim is proved.
Acknowledgments
This work was supported by the ERC NEMO grant, under the European Union’s
Horizon 2020 research and innovation programme, grant agreement number
788851 to INRIA. We also thank Mir-Omid Haji-Mirsadeghi for suggesting the
alternate proof of the last step of the proof of Theorem 1.1.
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