Exchangeable random measures

(1)

www.imstat.org/aihp 2015, Vol. 51, No. 3, 842–861

DOI:10.1214/13-AIHP584

Exchangeable random measures ¹

Tim Austin

Courant Institute, New York University, New York, NY 10012, USA. E-mail:tim@cims.nyu.edu Received 26 April 2013; revised 6 September 2013; accepted 19 September 2013

Abstract. LetAbe a standard Borel space, and consider the spaceA^N^(k) ofA-valued arrays indexed by all size-ksubsets ofN.

This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations ofN. The main result is a representation theorem for such “exchangeable” random measures, obtained using the classical representation theorems for exchangeable arrays due to de Finetti, Hoover, Aldous and Kallenberg.

After proving this representation, two applications of exchangeable random measures are given. The first is a short new proof of the Dovbysh–Sudakov Representation Theorem for exchangeable positive semi-definite matrices. The second is in the formulation of a natural class of limit objects for dilute mean-field spin glass models, retaining more information than just the limiting Gram–de Finetti matrix used in the study of the Sherrington–Kirkpatrick model.

Résumé. SoitAun espace de Borel standard, et soitA^N^(k)l’ensemble des tableaux à valeurs dansAindexés par les sous-ensembles deNde taillek. On s’intéresse aux mesures aléatoires sur un tel espace dont la loi est invariante par l’action naturelle des permutations deN. Le résultat principal est une représentation de ces mesures aléatoires « échangeables », obtenue à partir des théorèmes de représentation classiques de de Finetti, Hoover, Aldous et Kallenberg pour des tableaux échangeables.

Après avoir prouvé cette représentation, on en donne deux applications. La première est une nouvelle courte preuve du théorème de représentation de Dovbysh–Sudakov pour des matrices définies semi-positives échangeables. La seconde concerne la formulation d’une classe naturelle d’objets limites pour des modèles de champ moyen dilués pour des verres de spins qui capture plus d’information que la seule matrice limite de Gram–de Finetti qui est notamment utilisée dans l’étude du modèle de Sherrington–

Kirkpatrick.

MSC:Primary 60G09; secondary 60G57; 82B44

Keywords:Random measures; Exchangeability; Dovbysh–Sudakov Theorem; Dilute spin glass models

1. Introduction

The theory of exchangeable arrays of random variables emerged in work of Hoover [13,14], Aldous [1–3] and Kallen- berg [15,16], and amounts to a significant generalization of the classical de Finetti–Hewitt–Savage Theorem on exchangeable sequences. The heart of the theory is a collection of representation theorems for general such arrays, which then beget more specialized representation results such as the Dovbysh–Sudakov Theorem for exchangeable PSD matrices.

This note will consider the related setting of random measures on spaces of arrays, where now the laws of those random measures are assumed invariant under the relevant group action. Intuitively, this introduces an “extra layer of randomness.” In order to introduce these formally, let[n] := {1,2, . . . , n}forn∈N, letS_N=

n≥1S_[n]be the group

1Research partially supported by a research fellowship from the Clay Mathematics Institute.

(2)

of all permutations ofNwhich fix all but finitely many elements, and consider a measurable actionT: S_NEon a standard Borel spaceE. In full, this is a measurable function

T:S_N×E−→E:(σ, x)→T^σx such that

T^id^N=idE and T^σ¹T^σ²x=T^σ²^σ¹x ∀σ₁, σ₂, x.

As is standard, ifμ∈PrEandσ∈S_NthenT_∗^σμdenotes the image measure ofμunderT^σ.

Definition 1.1. IfEis a standard Borel space andT: S_NEis a measurable action,then anexchangeable random measure (ERM)on(E, T )is a random variableμtaking values inPrEsuch that

μ^law=T_∗^σμ ∀σ∈S_N; that is,

μ(A)

A⊆BorelE law=

μ

x: T^σx∈A

A⊆BorelE ∀σ∈S_N.

These are essentially what ergodic theorists call “quasi-factors” ([11], Chapter 8). We will study these for the group actions that underly the theory of exchangeable arrays. Given a standard Borel space Aandk∈N, the space ofk- dimensional arrays valued in AisA^N^(k), whereN^(k) denotes the set of size-ksubsets ofN. An element of such a space of arrays will often be denoted by(xe)_|e|=k or similarly. (In the following, one could focus instead on arrays indexed by orderedk-tuples, but we have chosen the symmetric case as it is a little simpler and arises more often in applications.) The groupS_Nacts onA^N^(k)by permuting coordinates in the obvious manner:

T^σ

(x_e)_|_e_|=_k

=(x_{σ (e)})_|_e_|=_k,

where σ (e)= {σ (i): i∈e}. Slightly more generally, our main results will also allow Cartesian products of such actions over finitely many differentk. Thus, our arrays will usually be indexed by the familyN⁽^≤^k)of subsets ofNof size at mostkfor some fixedk.

Examples. (1)If an exchangeable random measureμon(E, T )is deterministic,then its constant value must itself be invariant under the actionT.In caseE=A^N^(k)with the action above,this meansμis almost surely the law of an exchangeableA-valued,N^(k)-indexed array.

(2) On the other hand,ifμ is aT-invariant measure for any action (E, T ),then another way to obtain an ex- changeable random measure from it is to let

μ:=δX,

whereXis a random element ofEwith lawμ,andδXis the Dirac mass atX.

(3) In caseE=A^N^(k) with the action above,example(2)fits into a more general family as follows.The space of probability measuresPrAis also standard Borel with the Borel structure generated by evaluation of measures on Borel sets.Suppose(λe)_|_e_|=_k is an exchangeable array of(PrA)-valued random variables,and now let

μ=

|e|=k

λe.

This class of examples will feature again later.Such an example is called anexchangeable random product measure (ERPM).

(3)

(4) It is also easy to exhibit an ERM which is not ERPM.For example,let=(A, B) be a uniform random bipartition ofN(this is obviously exchangeable),and having chosenletμ∈Pr{0,1}^N⁽²⁾ be the probability which has two atoms of mass ¹₂on the points

1_A_N⁽²⁾ and 1_B_N⁽²⁾.

(5)Lastly,given a measurable family of exchangeable random measuresμ_t indexed by a parametert∈ [0,1),we may average over this parameter to obtain a mixture of these exchangeable random measures:

μ= ₁

0

μ_tdt.

This is clearly still exchangeable.

The main result of this paper characterizes all ERMs on spaces of arrays. To motivate it, we next recall the Repre- sentation Theorem for exchangeable arrays themselves. This requires some more notation.

First, for any setSwe letPSdenote the power set ofS.

Next, suppose thatB₀,B₁, . . . , B_kandAare standard Borel spaces. A Borel function f:B₀×B₁^k×B₂^[^k^]⁽²⁾× · · · ×B_k=

i≤k

B_i^[^k^]⁽ⁱ⁾−→A

ismiddle-symmetricif f

x, (x_i)_i_∈[_k_], (x_a)_a_∈[_k_](2), . . . , x_[_k_]

=f

x, (x_{σ (i)})_i_∈[_k_], (x_{σ (a)})_a_∈[_k_](2), . . . , x_[_k_] for allσ∈S_[_k_].

Given standard Borel spacesB0,B1, . . . , Bk andA0,A1, . . . , Ak, and middle-symmetric Borel functions f_i:

j≤i

B_j^[ⁱ^]^{(j )}−→A_i, i=0,1, . . . , k,

we will writeffor the function

i≤k

B_i^[^k^]⁽ⁱ⁾−→

i≤k

A^[_i^k^]⁽ⁱ⁾:(x_e)_e_⊆[_k_]→ f_|_e_|

(x_a)_a_⊆_e

e⊆[k], which combines all of thefi.

The tuple(f₀, . . . , f_k)is referred to as askew-product tuple, and the associated functionfas a function ofskew- product type; clearly the latter determines the former uniquely.

Example. Ifk=2,then a function of skew-product type[0,1)^P[²^]−→ [0,1)takes the form f (x, x ₁, x₂, x₁₂)=

f₀(x), f₁(x, x₁), f₁(x, x₂), f₂(x, x₁, x₂, x₁₂) .

It is easily checked that iffandgare functions of skew-product type for the samek, then so isg◦f. In terms of (f0, . . . , fk)and(g0, . . . , gk)this composition corresponds to the skew-product tuple

hi

(xa)a⊆[i] :=gi

f_|a| (xb)b⊆a

a⊆[i]

, i=0,1, . . . , k.

Slightly abusively, we will also writeffor the related function

i≤k

B_i^N⁽ⁱ⁾−→

i≤k

A^N_i ⁽ⁱ⁾:(x_e)_|_e_|≤_k→ f_|_e_|

(x_a)_a_⊆_e

|e|≤k, which also determines(f0, . . . , fk)uniquely.

(4)

Theorem 1.2 (Representation Theorem for Exchangeable Arrays; Theorem 7.22 in [18]). Suppose that A₀, A₁, . . . , A_k are standard Borel spaces and that (X_e)_|_e_|≤_k is an exchangeable random array of r.v.s with eachX_e valued inA_|_e_|.Then there are middle-symmetric Borel functions

f_i:[0,1)^P[ⁱ^]−→A_i, i=0,1, . . . , k, such that

(Xe)_|e|≤k law=

f_|e| (Ua)a⊆e

|e|≤k dfn=f

(Ue)_|e|≤k

,

where(U_e)_|_e_|≤_k is an i.i.d.family ofU[0,1)-r.v.s.

The companion Equivalence Theorem, which addresses the non-uniqueness of the representing functionf, will be recalled later.

To produce a random measure, the idea will simply be to use directing functionsfi that depend on two sources of randomness, and then condition on one of them.

Theorem A. Suppose thatμis an ERM onA0× · · · ×A^N_k^(k).Then there are middle-symmetric Borel functions f_i:

[0,1)× [0,1)_P[i]−→A_i such that

μ(·)^law=^P f

(U_e, V_e)_|_e_|≤_k

∈ ·|(U_e)_|_e_|≤_k ,

whereUeandVefore⊆N,|e| ≤kare all i.i.d.∼U[0,1).On the right-hand side,this is a measure-valued random variable as a function of the r.v.s(Ue)_|e|≤k.

We will find that after some manipulation of the problem, TheoremAcan be deduced from the Representation Theorem and Equivalence Theorems for exchangeable random arrays themselves.

The proof of TheoremAcan be considerably simplified whenk=1, so we will first prove that case separately. In that case, the structure given by TheoremAis essentially a combination of examples (3) and (5) above. To see this, we reformulate the result as follows.

Given a standard Borel spaceA, letB([0,1),PrA)denote the space of Lebesgue-a.e. equivalence classes of measurable functions[0,1)−→PrA. ThenB([0,1),PrA)has a natural measurable structure generated by the functionals

f → ₁

0

φ(t)f (t, B)dt

corresponding to all φ∈L^∞[0,1)and Borel subsets B⊆A. This measurable structure is also standard Borel: for instance, if one realizesAas a Borel subset of a compact metric space, then the above becomes the Borel structure of the topology of convergence in probability onB([0,1),PrA), which is Polish.

Theorem B. If μ is an ERM on A^N, then there is an exchangeable sequence of r.v.s (λi)_i_∈N taking values in B([0,1),PrA)such that

μ(·)^law= 1

0

i∈N

λi(t,·)

dt.

So whenk=1, every ERM is a mixture of ERPMs.

With the structure given by TheoremB, one may next apply the de Finetti–Hewitt–Savage Theorem to the sequence λi to obtain a random measureγ onB([0,1),PrA)such thatλi is obtained by first choosingγ and then choosingλi

(5)

i.i.d. with lawγ. We write Samp(γ)for the ERM obtained by this procedure, and refer toγ as adirecting random measureforμ.

After proving TheoremsA andB, we offer a couple of applications of the casek=1. These applications can also be given higher-dimensional extensions using the casesk≥2, but those extensions seem less natural. The reader interested only in the applications need not read the proof of the general case of TheoremA.

The first application is a new proof of the classicalDovbysh–Sudakov Theorem:

Dovbysh–Sudakov Theorem. Suppose(R_ij)_i,j_∈N is a random matrix which is a.s.positive semi-definite,and is exchangeable in the sense that

(Rσ (i)σ (j ))_i,j ^law=(R_ij)_i,j ∀σ∈S_N.

Then there are a separable real Hilbert spaceHand an exchangeable sequence(ξ_i, a_i)_i_∈Nof random variables valued inH× [0,∞)such that

(R_ij)_i,j ^law=

ξ_i, ξ_j +δ_ija_i

i,j, whereδij is the Kronecker delta.

This first appeared in [9], and more complete accounts were given in [12] and [21]. The proofs of Hestir and Panchenko start with the Aldous–Hoover Representation Theorem, which treats(Rij)i,j as a general two-dimensional exchangeable array. They then require several further steps to show that the PSD assumption implies a simplification of that general Aldous–Hoover representation into the form promised above. On the other hand, we will find that if one simply interprets(R_ij)_i,j as the covariance matrix of an exchangeable random measure, then one can read off the Dovbysh–Sudakov Theoremfrom Theorem B, which in turn does not require the Aldous–Hoover Theorem.

Our second application is to the study of certain mean-field spin glass models, and particularly Viana and Bray’s dilute version of the Sherrington–Kirkpatrick model [25]. In the case of the original Sherrington–Kirkpatrick model a great deal has now been proven, much of it relying on the notions of “random overlap structures” and their directing random Hilbert space measures: see, for instance, Panchenko’s monograph [22]. The analogous theory for dilute models is less advanced. In this note we will simply sketch how the main conjecture of Replica Symmetry Breaking can be formulated quite neatly in terms of limits of exchangeable random measures, translating from the earlier works [20,23]. We will not recall most of the spin glass theory behind this conjecture, but will refer the reader to those references for more background.

2. The replica trick

The key to TheoremAis the simple observation that the law of a random measure on some spaceEcan be equivalently described by the law of a random sequence inE, obtained by first sampling and quenching that random measure, and then sampling i.i.d. from it. This idea is standard in the more general setting of representing quasi-factors in ergodic theory ([11], Chapter 8). In a sense, it is an abstract version of the “replica trick” from the statistical physics of spin glasses ([19]). In physics, the phrase “replica trick” usually refers to the calculation of the sequence of moments of the (random) partition function of a random Gibbs measure, which is then fed into an ansatz for guessing more about the law of the partition function, such as the expected free energy. This resembles our “replica trick” insofar as computing a moment of the random partition function amounts to computing the partition function for the law of several i.i.d.

samples from the random Gibbs measure.

Before we proceed, first observe that, since any standard Borel space is isomorphic to a Borel subset of a compact metric space, we may replace the spacesA₀, . . . , A_k with such enveloping compact spaces in TheoremsAandB, and so assume these spaces are themselves compact. We will make this assumption throughout the proofs of those theorems, although some non-compact examples will re-appear later in the applications.

Proposition 2.1 (Replica trick). Ifμis an ERM on

i≤kA^N_i ⁽ⁱ⁾,then there are auxiliary standard Borel spacesA0, A1, . . . , Akand an exchangeable array(Ye, Xe)_|e|≤k of random variables such that

(6)

• each(Y_e, X_e)takes values inA_|_e_|×A_|_e_|,and

• one has μ(·)^law=^P

(X_e)_|_e_|≤_k∈ ·|(Y_e)_|_e_|≤_k .

Proof. After enlarging the background probability space if necessary, we may couple the random variableμwith a doubly-indexed family of random variables

(Xi,e)_i_∈N_,e_∈N(≤k), (Xe)_e_∈N(≤k)

, (1)

all taking values in one of theA_i’s, as follows:

• first, sample the random measureμitself;

• then, choose the sub-families(Xe)_|e|≤k,(X1,e)_|e|≤k,(X2,e)_|e|≤k, . . .independently with lawμ.

In notation, this coupling is defined by P

(Xe)_|e|≤k∈da, (X1,e)_|e|≤k∈da1, (X2,e)_|e|≤k∈da2, . . .|μ

=μ(da)·μ(da1)·μ(da2)· · · ·.

Having done this, letA_i:=A^N_i and letY_e:=(X_j,e)_j_∈N∈A_|_e_|for eache∈N⁽^≤^k). The exchangeability ofμimplies that the joint distribution of the family (1) is invariant under applying elements ofS_Nto the indexing setse, and hence that the process(Y_e, X_e)_|_e_|≤_k is exchangeable.

On the other hand, since we assume eachA_i is compact, so is

i≤kA^N_i⁽ⁱ⁾, and now the Law or Large Numbers shows that in the above process one has the a.s. convergence of empirical measures

1 N

N

n=1

δ_(X_n,e₎_|e|≤k−→μ

in the vague topology on Pr

i≤kA^N_i⁽ⁱ⁾. Therefore in the process

(Y_e)_|_e_|≤_k, (X_e)_|_e_|≤_k,μ

the family of r.v.s(Y_e)_|_e_|≤_k determineμa.s., whereas conditionally onμthe family(Y_e)_|_e_|≤_k becomes independent from(X_e)_|_e_|≤_k. This implies that

P

(X_e)_|_e_|≤_k∈ ·|(Y_e)_|_e_|≤_k

=^P

(X_e)_|_e_|≤_k∈ ·|μ

=μ(·) a.s.,

as required.

3. Proofs in one dimension

3.1. Some preliminaries

We will repeatedly need the following standard tool from measure-theoretic probability. See, for instance, the slightly- stronger Theorem 6.10 in [17].

Lemma 3.1 (Noise-Outsourcing Lemma). Suppose thatAandB are standard Borel spaces and that(X, Y )is an (A×B)-valued r.v.Then,possibly after enlarging the background probability space,there are a r.v.U ∼U[0,1) coupled withXandY and a Borel functionf:A× [0,1)−→Y such thatUis independent fromXand

(X, Y )=

X, f (X, U ) a.s.

(7)

Of course, the functionf in this lemma is highly non-unique. The degenerate case in whichXis deterministic is still important: it reduces to the assertion that for any standard Borel probability space(B, ν)there is a Borel function f:[0,1)−→Bsuch thatf (U )∼νwhenU∼U[0,1).

Finally, let us recall the full de Finetti–Hewitt–Savage Theorem for the casek=1, which is rather stronger than just the casek=1 of Theorem1.2. The following is the combination of Proposition 1.4 and Corollaries 1.5 and 1.6 in [18].

Theorem 3.2. SupposeAis a compact metric space and(X_n)_nis an exchangeable sequence ofA-valued r.v.s.Then the sequence of empirical distributions

W_N:= 1 N

N

n=1

δ_X_n∈PrA

converges a.s.to a(PrA)-valued r.v.Wwhich has the following properties:

(i) Wis a.s.a function of(X_n)_n,

(ii) Wgenerates the tailσ-algebra of(X_n)_nup toμ-negligible sets;

(iii) the r.v.sXnare conditionally i.i.d.givenW;

(iv) ifZis any other r.v.on the same probability space such that (Z, X₁, X₂, . . .)^law=(Z, X_{σ (1)}, X_{σ (2)}, . . .) ∀σ∈S_N, thenZis conditionally independent from(Xn)noverW. 3.2. Proofs in one dimension

Proof of TheoremAin one dimension. Supposeμis an ERM onA^N and let(Y_n, X_n)_nbe a process as given by Proposition2.1.

We next apply Theorem3.2twice: first to the sequence(Y_n)_n, to obtain a r.v.W taking values inE:=PrA; and secondly to(Y_n, X_n)_n, to obtain a r.v.Ztaking values inF :=Pr(A×A). From their definitions as limits of empirical distributions,W is almost surely a function ofZ. On the other hand, property (iv) of Theorem3.2gives thatZ is conditionally independent from(Y_n)_noverW.

Now pick ann∈N. By Lemma3.1, there is a Borel functionf1:E×F×A× [0,1)−→Asuch that (W, Z, Y_n, X_n)^law=

W, Z, Y_n, f₁(W, Z, Y_n, V_n) ,

where(V_n)_nare i.i.d.∼U[0,1)and are independent from(W, Z, Y_n). Moreover, this samef₁works for everyn, by exchangeability. It follows that in fact

W, Z, (Y_n, X_n)_n_∈Nlaw

= W, Z,

Y_n, f₁(W, Z, Y_n, V_n)

n∈N

, (2) because both sides have the same marginals for individualn, and both sides are conditionally i.i.d. over(W, Z), so all finite-dimensional marginals agree.

Next, another appeal to Lemma3.1gives a Borel functiong:E× [0,1)−→F such that (W, Z)^law=

W, g(W, V )

with a new independentV ∼U[0,1). This implies that W, Z, (Y_n)_n_∈Nlaw

=

W, g(W, V ), (Y_n)_n_∈N ,

becauseZis conditionally independent from(Yn)noverW, so again all finite-dimensional marginals agree. Combin- ing this with (2) gives

W, Z, (Yn, Xn)n∈Nlaw

=

W, g(W, V ),

Yn, f2(W, Yn, V , Vn)

n∈N

,

(8)

where f2

w, y, v, v :=f1

w, g(w, v), y, v . It follows that

P

(X_n)_n∈ ·|(Y_n)_n

=^P

f₂(W, Y_n, V , V_n)

n∈ ·|(Y_n)_n .

Finally, we may apply de Finetti’s Theorem again, this time to(Yn)n, to obtain Borel functionsh1:[0,1)−→Eand h₂:[0,1)²−→Asuch that

W, (Y_n)_n_∈Nlaw

=

h₁(U ),

h₂(U, U_n)

n∈N

,

whereUand(U_n)_nare i.i.d.∼U[0,1)r.v.s, independent of everything else. Letting f

u, u, v, v :=f₂

h₁(u), h₂ u, u

, v, v , substituting for(W, (Y_n)_n)in the above gives

P

(X_n)_n∈ ·|(Y_n)_nlaw

=^P

f (U, U_n, V , V_n)

n∈ ·|U, (U_n)_n ,

as required.

Proof of Theorem B. By the Law of Iterated Conditional Expectation, the representation obtained above may be re-written as

μ(·)^law=^E P

f (U, Un, V , Vn)

n∈ ·U, (Un)n, V

|U, (Un)n

=^E

n∈N P

f (U, Un, V , Vn)∈ ·U, Un, V

|U, (Un)n

= ₁

0

n∈N

λn(t,·)

dt,

where

λn(t,·):=^P

f (U, Un, t, Vn)∈ ·|U, Un

.

As functions of(U, U_n)for eachn, these form an exchangeable sequence of r.v.s taking values inB([0,1),PrA), so

the proof is complete.

3.3. Relation to row–column exchangeability

A relative of exchangeability for a two-dimensional random array(X_i,n)_(i,n)_∈N2isrow–column exchangeability, which asserts that

(Xσ (i),τ (n))_i,n^law=(X_i,n)_i,n ∀σ, τ∈S_N.

Sinceσ andτ may be chosen separately, this is a rather stronger symmetry than ordinary two-dimensional exchangeability. Here, too, there is a representation theorem due to Aldous and Hoover, and also a version in arbitrary dimensions due to Kallenberg, who calls such arrays “separately exchangeable.”

(9)

Theorem 3.3 (Corollary 7.23 in [18]). If(X_i,n)_i,nis anA-valued row–column exchangeable array then there is a Borel function[0,1)⁴−→Asuch that

(X_i,n)_i,n^law=

f (Z, U_i, V_n, W_i,n)

i,n,

whereZ,U_i fori∈N,V_nforn∈NandW_i,nfor(i, n)∈N²are i.i.d.∼U[0,1).

An alternative proof of TheoremBcan be given via Theorem3.3. One begins with the construction of the two- dimensional random array(Xi,n)i,nas in the proof of Proposition2.1(where the setsehave become singletonsn).

Since this array is row–column exchangeable, the Representation Theorem gives (X_i,n)_i,n^law=

f (U, U_i, V_n, W_i,n)

i,n

for some Borel directing functionf:[0,1)⁴−→A, whereU,U_ifori∈N,V_nforn∈NandW_i,nfori, n∈Nare i.i.d.

∼U[0,1). One can now read off a directing random measureγ (U )onB([0,1),PrA), a function ofU∼U[0,1), in the following two steps:

• first, for each fixedUandUone obtains an elementλ(U, U)∈B([0,1),PrA)according to λ

U, U

(t,·)=^PW

f

U, U, t, W

∈ ·

, W∼U[0,1);

• second,γ (U )is the distribution ofλ(U, U)whereU∼U[0,1).

On the other hand, a couple of simple applications of the Noise-Outsourcing Lemma show that any directing random measureγ onB([0,1),PrA)can be represented this way, so this gives a bijective correspondence

directing random measures onB

[0,1),PrA

up to equivalence

↔

directing functions[0,1)⁴−→Aup to equivalence ,

where “up to equivalence” refers to the possibility that different directing random measures or directing functions may give rise to the same row–column exchangeable array.

This approach is the basis of the paper [20], to be discussed later. It is quick, but at the expense of assuming Theorem3.3.

On the other hand, our approach to TheoremBdoes not use any exchangeability theory in dimensions greater than one. Moreover, one can reverse the idea above to give a fairly quick proof of Theorem3.3using TheoremB.

Proof of Theorem3.3from TheoremB. First let(Xi,n)_i,n=((X_i,n)_i)_n, thought of as an exchangeable sequence of A^N-valued r.v.s. By the de Finetti–Hewitt–Savage Theorem applied to the exchangeability inn, its law is a mixture of product measures; equivalently, there is a(PrA^N)-valued r.v.μsuch that

law (Xi,n)i

n

=^E μ^⊗N

.

On the other hand, for anyσ∈S_Nthe exchangeability inigives E

μ^⊗N

=law (X_i,n)_i

n

=law

(X_{σ (i),n})_i

n

=^E

T_∗^σμ_⊗N ,

where T^σ:A^N −→A^N is the corresponding coordinate-permuting transformation. By the uniqueness of the de Finetti–Hewitt–Savage decomposition, this implies

μ^law=T_∗^σμ ∀σ∈S_N,

soμis an ERM. Therefore TheoremBgives law

(X_i,n)_i

n

=^E 1

0

i∈N

λi(t,·)dt _⊗N

(10)

for some exchangeable random sequence(λi)_i taking values inB([0,1),PrA).

Next, applying the de Finetti–Hewitt–Savage Theorem to the sequence (λi)i itself gives a Borel function F:[0,1)²−→B([0,1),PrA)such that the above becomes

law (X_i,n)_i

n

=^EZ,(Ui)i

₁

0

i∈N

F (Z, U_i, t,·)dt _⊗N

=^EZ,(Ui)i,(Vn)n

(i,n)∈N²

F (Z, U_i, V_n,·), (3)

whereZ,U_i for i∈NandV_n for n∈Nare i.i.d.∼U[0,1), and in the second equality we have simply changed notation from “₁

0·dt” to “E_V_n.”

Finally, by Lemma3.1there is a Borel functionf:[0,1)⁴−→Asuch that P

f (Z, U, V , W )∈ ·|Z, U, V

=F (Z, U, V ,·) a.s.

whenZ,U,V,W are i.i.d.∼U[0,1), and now the right-hand side of (3) becomes law((f (U, Ui, Vn, Wi,n)i)n), as

required.

4. Proof in higher dimensions

Not all parts of Theorem 3.2generalize to higher-dimensional arrays, and instead we must make a more careful argument using the Equivalence Theorem4.1below.

4.1. Some more preliminaries

The Equivalence Theorem characterizes when two functions direct the same process in the setting of Theorem1.2. Its formulation needs the following notion. LetC= [0,1)^r andD= [0,1)^sfor somer, s∈N. Then a skew-product tuple (f₀, . . . , f_k)in which eachf_i:

j≤iC^[ⁱ^]^{(j )}−→Dfor eachigives rise to a skew-product-type functionf:C^P[^k^]−→

D^P[^k^], which is a map between Euclidean cubes. We will write that the skew-product tuple isLebesgue-measure- preservingif for alli=0,1, . . . , kand all(xa)_a_[_i_]∈C^P[ⁱ^]\[ⁱ^], one has

U∼U(C) ⇒ f_i

(x_a)_a_[_i_], U

∼U(D).

This implies, in particular, thatfpushes Lebesgue measure onC^P[^k^]to Lebesgue measure onD^P[^k^](although these cubes have different dimensions ifr=s). However, the assertion that the skew-product tuple is Lebesgue-measure- preserving can be strictly stronger than this, in case the functionsf_i are not all injective.

The Equivalence Theorem is as follows.

Theorem 4.1 (Equivalence Theorem for directing functions; Theorem 7.28 in [18]). Iff ,f:[0,1)^P[^k^]−→A₀×

· · · ×A^[_i^k^]⁽ⁱ⁾× · · · ×A_kare functions of skew-product type such that f_|_e_|

(U_a)_a_⊆_e

|e|≤k law=

f_|_e_|

(U_a)_a_⊆_e

|e|≤k,

then there are functions G, G:[0,1)^P[^k^] −→ [0,1)^P[^k^] of skew-product type, whose skew-product tuples are Lebesgue-measure-preserving,and which make the following diagram commute:

(11)

[0,1)^P[^k^]

G G

[0,1)^P[^k^]

f

[0,1)^P[^k^]

f

A0×A^k₁× · · · ×Ak

In connection with this theorem, we will also need the following “factorization” result.

Corollary 4.2. LetU_⊆[_k_]=(U_e)_e_⊆[_k_]andV_⊆[_k_]be independent uniform r.v.s valued in[0,1)^P[^k^].If G:[0,1)^P[^k^]−→ [0,1)^P[^k^]

is a function of skew-product type whose skew-product tuple is Lebesgue-measure-preserving,then there is another function

H:

[0,1)× [0,1)_P[k]−→ [0,1)^P[^k^]

of skew-product type,whose skew-product tuple is Lebesgue-measure-preserving,and such that U_⊆[_k_]=G

H (U_⊆[_k_], V_⊆[_k_]) a.s.

Another way to express this is that the maps in the following diagram come from Lebesgue-measure-preserving skew-product tuples and a.s. commute:

([0,1)× [0,1))^P[^k^]

Π

H [0,1)^P[^k^]

G

[0,1)^P[^k^] where

Π

(x_e, y_e)_e_⊆[_k_]

=(x_e)_e_⊆[_k_] is the obvious projection.

Geometrically, the intuition here is thatGis “almost onto” (since its image measure is Lebesgue), and that as a result one can represent it as the projection mapΠ after usingH to “straighten out the fibres.”

Proof of Corollary4.2. Let Gbe defined by the skew-product tuple (G₀, . . . , G_k). We must construct the skew- product tuple(H0, . . . , Hk)that definesH. In terms of these tuples, our requirement is that

G_i H_|_e_|

(U_a, V_a)_a_⊆_e

e⊆[i]

=U_[_i_] a.s.∀i=0,1, . . . , k. (4)

Wheni=0 this simplifies to G0

H0(U0, V0)

=U0 a.s.

We can obtain such anH₀from the Noise-Outsourcing Lemma3.1as follows. LetZ₀be a U[0,1)-r.v. and letX₀:=

G0(Z0), so this is also ∼U[0,1). Applying Lemma 3.1to the pair (X0, Z0) gives a Borel function H0:[0,1)× [0,1)−→ [0,1)such that

(X0, Z0)=

X0, H0(X0, Y0) a.s.

(12)

for someY₀∼U[0,1)independent fromX₀. SinceX₀=G₀(Z₀), applyingG₀to the second coordinates here gives X₀=G₀

H₀(X₀, Y₀) a.s.

The general case now follows by induction oni. Suppose thati≥1, letY_e fore[i]andZ_efore⊆ [i]be i.i.d.

∼U[0,1); defineX_e:=G_|_e_|((Z_a)_a_⊆_e)for alle⊆ [i]; and assume thatH₀, . . . , H_i₋₁have already been constructed such thatZ_e:=H_|_e_|((X_a, Y_a)_a_⊆_e)for eache[i]. Applying Lemma3.1again gives a Borel functionH_i:([0,1)× [0,1))^P[ⁱ^]−→ [0,1)and a r.v.Y_[_i_]∼U[0,1)such that

(X_e)_e_⊆[_i_], (Y_e)_e_[_i_], Z_[_i_]

=

(X_e)_e_⊆[_i_], (Y_e)_e_[_i_], H_i

(X_e)_e_⊆[_i_], (Y_e)_e_⊆[_i_] ,

and as before this is equivalent to the desired equality (4).

4.2. Completion of the proof

We need the following enhancement of Proposition2.1.

Lemma 4.3. Ifμis an ERM on

i≤kA^N_i⁽ⁱ⁾,then there is an exchangeable array(U_e, X_e)_|_e_|≤_ksuch that

• (U_e)_|_e_|≤_kare i.i.d.∼U[0,1),

• eachX_etakes values inA_|_e_|,

• and one has μ(·)^law=^P

(Xe)_|e|≤k∈ ·|(Ue)_|e|≤k

.

Proof. Let(Y_e, X_e)_|_e_|≤_k be the process given by Proposition2.1, with each(Y_e, X_e)taking values inA_|_e_|×A_|_e_|. By the Representation Theorem1.2applied to(Y_e)_|_e_|≤_k, there is a functionf:[0,1)^P[^k^]−→

i≤kA^[^k^]

(i)

i of skew-product type such that

(Ye)_|e|≤klaw

=f

(Ue)_|e|≤k

,

where(Ue)_|e|≤kis an i.i.d.∼U[0,1)array.

Now consider the coupling (Ue, Xe)_|e|≤k whose law is the relatively independent product over the condition (Ye)_|e|≤k=f ((U e)_|e|≤k):

P

(Ue)e∈du, (Xe)e∈da

=^P

(U_e)_e∈du

·^P

(X_e)_e∈da|(Y_e)_e=f (u)

. (5)

This is exchangeable and has the three desired properties. The exchangeability follows because both factors on the right-hand side of (5) are invariant under the action ofS_Non the indexing sete, by the exchangeability of(U_e)_eand (Y_e, X_e)_e. The first two of the properties listed are obvious, and the third follows from Proposition2.1because the above relative product formula gives

P

(X_e)_e∈da|(U_e)_e=u

=^P

(X_e)_e∈da|(Y_e)_e=f (u) ,

and we conditioned on the equality(Y_e)_e=f ((U _e)_e).

Proof of TheoremA. Let the process(U_e, X_e)_|_e_|≤_k be as in the preceding corollary. Applying Theorem1.2to this whole process gives functions g:[0,1)^P[^k^]−→ [0,1)^P[^k^] andh:[0,1)^P[^k^]−→

i≤kA^[_i^k^]⁽ⁱ⁾ of skew-product type such that

(U_e)_|_e_|≤_k, (X_e)_|_e_|≤_klaw

= g

U_e

|e|≤k

,h U_e

|e|≤k

, (6)

Exchangeable random measures

Exchangeable random measures 1

Tim Austin

Exchangeable random measures ¹