Large deviations for a triangular array of exchangeable random variables

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LARGE DEVIATIONS FOR A TRIANGULAR ARRAY OF EXCHANGEABLE RANDOM VARIABLES

GRANDES DÉVIATIONS POUR UN TABLEAU TRIANGULAIRE DE VARIABLES ALÉATOIRES

ÉCHANGEABLES

José TRASHORRAS

Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 7, UFR de Mathématiques, case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France

Received 20 September 2000, revised 3 April 2001

ABSTRACT. – In this paper we consider a triangular array whose rows are composed of finite exchangeable random variables. We prove that, under suitable conditions, the sequence defined by the empirical measure process of each row satisfies a large deviation principle. We first study the particular case where the rows are given by sampling without replacement from fixed urns. Then we prove a large deviation principle in the general setting, by identifying finite exchangeable random variables and sampling without replacement from urns with random composition.2002 Éditions scientifiques et médicales Elsevier SAS

AMS classification: Primary 60F10; secondary 60G09 and 62G09

Keywords: Large deviations; Exchangeable random variables; Sampling without replacement RÉSUMÉ. – Nous considérons un tableau triangulaire dont les lignes sont composées de variables aléatoires fini-échangeables. Nous prouvons sous certaines conditions que la suite définie par le processus de mesure empirique de chaque ligne vérifie un principe de grandes déviations. Dans un premier temps nous traitons le cas particulier où chaque ligne résulte du tirage sans remise dans une urne de composition donnée. Nous en déduisons ensuite un principe de grandes déviations dans le cas général, en identifiant les variables aléatoires fini-échangeables avec le tirage sans remise dans des urnes de composition aléatoire.2002 Éditions scientifiques et médicales Elsevier SAS

E-mail address: Jose.Trashorras@gauss.math.jussieu.fr (J. Trashorras).

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1. Introduction

We say that a sequence of Borel probability measures(Pⁿ)n∈Non a topological space obeys a Large Deviation Principle (hereafter abbreviated LDP) with rate functionI and in the scale (an)n∈N if(an)n∈N is a real-valued sequence satisfyingan→ ∞and I is a non-negative, lower semicontinuous function such that

− inf

x∈A^oI (x)lim inf

n→∞

1 an

logPⁿ(A)lim sup

n→∞

1 an

logPⁿ(A)−inf

x∈ ¯A

I (x)

for any measurable set A, whose interior is denoted by A^o and closure by A. Unless¯ explicitly stated otherwise, we will take an =n. If the level sets {x: I (x) α} are compact for every α <∞, I is called a good rate function. With a slight abuse of language we say that a sequence of random variables obeys a LDP when the sequence of measures induced by these random variables obeys a LDP. For a background on the theory of large deviations, see Dembo and Zeitouni [6] and references therein.

In this paper, we are interested in the LD behavior of finite exchangeable random variables. The word exchangeable appears in the literature for both infinite exchangeable sequences of random variables, and finite exchangeable random vectors. A sequence of random variables (X1, . . . , Xn, . . .)defined on a probability space(,A,P) is infinite exchangeable if and only if for every permutation τ onNsuch that|{i, τ (i)=i}|<∞ the following identity in distribution holds

(X1, . . . , Xn, . . .)=^D(Xτ (1), . . . , Xτ (n), . . .).

An n-tuple(X1, . . . , Xn)of random variables defined on the same probability space is finite exchangeable or n-exchangeable (to indicate the number of random variables) if and only if for all permutations σ on{1, . . . , n}it satisfies the identity in distribution

(X1, . . . , Xn)=^D(Xσ (1), . . . , Xσ (n)).

Finite and infinite exchangeability are related since any n-tuple extracted from an infinite exchangeable sequence of random variables is n-exchangeable. While LD for infinite exchangeable sequences have been entirely studied by Dinwoodie and Zabell [9], much less is known in the more intricate case of finite exchangeable random variables.

After introducing our setting, we shortly review below known facts about exchangeable random variables. We refer to Aldous [1] for a large survey on this topic.

Throughout the sequel(, d)will denote a Polish space, andM⁺()[resp. M¹()]

the space of Borel non-negative measures [resp. probability measures] on . These spaces will always be equipped with the topology of weak convergence, and we shall denote convergence in this topology by µ^{n w}→µ. Let us recall that the dual-bounded- Lipschitz metric β on M⁺() is compatible with this topology (see Dembo and Zajic [4], Appendix A.1).

De Finetti’s well-known theorem (see, for example, [12]) states that any valued infinite exchangeable sequence of random variables (X1, . . . , Xn, . . .) defined on

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(,A,P)is a mixture of independent and identically distributed sequences of random variables, i.e. for any Borel setAofⁿ

P(X1, . . . , Xn)∈A=

Pθ

(X1, . . . , Xn)∈Aγ (dθ ),

whereγ is a probability measure on a closed subsetofM¹(), and for everyθ ∈, Pθis a probability measure defined on(,A)such thatX1, . . . , Xn, . . .are independent and identically distributed under Pθ. Using this result, Dinwoodie and Zabell [9] have shown that ifis compact, the distribution of ¹_nⁿ_i₌₁δXi underPsatisfies a LDP with good rate function

I (ν)=inf

θ∈H (ν|πθ),

whereπθ=Pθ◦X₁⁻¹andH (· | ·)stands for the usual relative entropy (see Dupuis and Ellis [10] for a nice account on relative entropy).

Nevertheless, de Finetti’s theorem is not valid for finite exchangeable random variables, as can be seen in the following simple example that arises in sampling.

Consider an urn with nlabelled balls (x1, . . . , xn). The result (X1, . . . , Xn) ofndraws without replacement among(x1, . . . , xn)is ann-exchangeable random vector that cannot be represented as a mixture of independent and identically distributed random variables.

In this special case, Dembo and Zeitouni [5] have showed that if ¹_nⁿ_i₌₁δx_i

→w µthen, for fixedt0∈ ]0,1[,the distribution of_[_nt¹

0]

_[nt₀]

i=1 δXi follows a LDP in the scale[nt0]and with good rate function

I (ν, t0, µ)=

H (ν|µ)+⁽¹⁻_t^t⁰⁾

0 H^µ₁⁻₋^t_t⁰^ν

0 |µ if ^µ₁⁻₋^t_t⁰^ν

0 ∈M¹(),

∞ otherwise.

Another well-known fact is that a family of n-exchangeable random variables can be approximated by n independent and identically distributed random variables in the variation norm (see Diaconis and Freedman [8]). However, this property does not give any hint for the LDP.

Here we consider a finite exchangeable triangular array ((X_iⁿ)1in)n∈N of valued random variables defined on (,A,P), i.e., each row (X₁ⁿ, . . . , Xⁿ_n) is finite exchangeable. We define the associated sequence of empirical measure processes by

Lⁿ_t = 1 n

[nt]

i=1

δXⁿ_i (1)

for every t∈ [0,1]. The process(Lⁿ_t)t∈[0,1]belongs to the spaceD[[0,1], (M⁺(), β)] of all maps defined on[0,1]that are continuous from the right and have left limits. This space is endowed with the topology defined by the uniform metric

β_∞(y_·, z_·)= sup

t∈[0,1]β(yt, zt), (2)

wherey_·is a shortcut for(yt)t∈[0,1].

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The experience we are interested in can be heuristically described this way: From any n-tuple(Y_iⁿ)1in of random variables one can simply obtain an n-exchangeable random vector(Xⁿ_i)1inby sampling without replacement from an urn withnlabelled balls (Y₁ⁿ, . . . , Y_nⁿ). Equivalently, we let in this case X_iⁿ=Y_{σ (i)}ⁿ , for i =1, . . . , n, with σ =σⁿ a random permutation on{1, . . . , n}which is independent from(Y_iⁿ)1inand uniformly distributed. Our purpose in this paper is to derive the LDP for (Lⁿ_t)t∈[0,1]

from the LDP for ¹_nⁿ_i₌₁δY_iⁿ. Now, let us describe our setting rigorously. Let Bⁿ

be the Borel σ-algebra on ⁿ and Pⁿ be any probability measure on (ⁿ,Bⁿ). We denote by (Y₁ⁿ, . . . , Y_nⁿ) the coordinate maps on (ⁿ,Bⁿ) when we consider them distributed according toPⁿ. LetPⁿbe the probability measure defined on every product A1× · · · ×Anof measurable subsets ofby

Pⁿ(A1× · · · ×An)= 1 n!

σ∈Sn

Pⁿ(Aσ (1)× · · · ×Aσ (n)), (3) whereSn is the symmetric group of ordern. We denote by(X₁ⁿ, . . . , Xⁿ_n)the coordinate maps on (ⁿ,Bⁿ)when its joint law is Pⁿ. Clearly, the random variables (X_iⁿ)₁in

are n-exchangeable. Let(,A,P)be the probability space associated to the sequence ((ⁿ,Bⁿ,Pⁿ))n∈N. Note that the mapping from ⁿ to D[[0,1], (M⁺(), β)] defined by (Lⁿ_t)t∈[0,1]is continuous, hence Borel measurable. As mentioned before, our goal is to derive the LDP [resp. the weak law of large numbers] for the distribution of(Lⁿ_t)t∈[0,1]

under Pⁿ from the LDP [resp. the weak law of large numbers] for the distribution of

1 n

n

i=1δY_iⁿunderPⁿ. Remark that [9] does not apply in this case.

The key to the proof is the following elementary fact. The law of (X₁ⁿ, . . . , Xⁿ_n) conditioned on {¹_nⁿi=1δXⁿ_i =ρ}, where ρ is an atomic measure whose atoms weigh

k

n (1kn), is the law of sampling without replacement among these atoms counted with their frequency of appearance in ρ. Hence our analysis essentially reduces to the following particular case. Let((y_iⁿ)1in)n∈N be a fixed triangular array of elements of , whose composition is given by(µⁿ=_n¹ⁿi=1δy_iⁿ)n∈N, possibly with ties. For every n∈N, we sample without replacement from the urn containing(y_iⁿ)₁inand we denote by xⁿ_i the ith element drawn. We call Pⁿ(· ;µⁿ) the distribution on ⁿ related to this sampling. For every n∈Nit clearly makes(x_iⁿ)1ina finite exchangeable vector. For allt∈ [0,1]we set

l_tⁿ=1 n

[nt]

i=1

δxⁿ_i, (4)

and for allµ∈M¹()we letACµ be the space of all mapsνt:[0,1] →M⁺()such that:

1. νt−νs∈M⁺()is of total masst−s for all 0st1.

2. ν0=0 andν1=µ.

3. ν_·possesses a weak derivative for almost everyt∈ [0,1]. We call weak derivative the limit

˙ νt =lim

ε→0

νt+ε−νt

ε , (5)

provided this sequence converges inM¹().

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In the sequel, by distribution of (l_tⁿ)t∈[0,1] we will mean its distribution under the probability measure Pⁿ(· ;µⁿ). It is an abuse of language, but there cannot be any confusion since the triangular array ((y_iⁿ)1in)n∈N is fixed. Our first result is the following.

THEOREM 1. – Ifµ^{n w}→µthen(l_tⁿ)t∈[0,1]obeys a LDP onD[[0,1], (M⁺(), β)]with good rate function

I_∞(ν_·, µ)= ⁰¹H (ν˙s|µ)ds ifν.∈ACµ,

∞ elsewhere. (6)

Theorem 1 can be viewed as a LDP for the so-called microcanonical distributions.

Simple microcanonical distributions are obtained from independent and identically distributed random variables X1, . . . , Xn by conditioning on the value of a functional of their empirical measure. The question of interest is then whether or not there is convergence of the marginal distribution ofX1under the conditional probability, when n→ ∞. For general background concerning microcanonical distributions we refer to Stroock and Zeitouni [18]. What we prove here is a LD result for the distribution of the contraction(Lⁿ_t =¹_n^[i^nt=1^]δX_i)t∈[0,1]ofX1, . . . , Xn, when these random variables are n-exchangeable, under a strong conditioning.

Next, taking into account the fluctuations of the compositionµⁿ of the urn, we obtain in this case a more involved result. LetQⁿbe the distribution ofLⁿ₁=_n¹ⁿi=1δXⁿ_i under Pⁿ. Note that this probability measure on M¹()is also the distribution of ¹_nⁿ_i₌₁δY_iⁿ

under Pⁿ. Let M^1,n() be the subset of M¹() composed of all atomic measures

1 n

_n

i=1δxi for(x1, . . . , xn)∈ⁿpossibly with ties, andAC= µ∈M¹()ACµ. Since PⁿLⁿ_· ∈A=

M^1,n()

Pⁿl_·ⁿ∈A;ρQⁿ(dρ) (7)

for every borelian AofD[[0,1], (M⁺(), β)], Theorem 1 tells us that (Lⁿ_t)t∈[0,1] is a mixture of Large Deviation Systems (from now on abbreviated LDS), in the sense of Dawson and Gartner [3]. Hence, the announced LDP holds by virtue of a result due to Grunwald [13].

THEOREM 2. – Suppose thatLⁿ₁ follows a LDP on M¹()with good rate function J. Then(Lⁿ_t)t∈[0,1]follows a LDP onD[[0,1], (M⁺(), β)]with good rate function

I (ν_·)=I_∞(ν_·, ν1)+J (ν1)= ⁰¹H (ν˙s|ν1)ds+J (ν1) ifν_·∈AC,

∞ elsewhere. (8)

Even in the simple case of binary valued finite exchangeable random variables there is no general result concerning the LD behavior ofLⁿ₁. So Theorem 2 seems to be the best result that can be stated in this setting.

The paper is organized as follows. In Section 2 we consider a fixed triangular array ((y_iⁿ)1in)n∈N of elements of. Generalizing a technique from [5], we prove that if

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µⁿ = ¹_nⁿ_i₌₁δyⁿ_i

→w µ we have a LDP for (l_tⁿ₀, . . . , l_tⁿ_d) on M⁺()^d⁺¹, for all d ∈N^∗ and all strictly ordered (d+1)-tuples t=(t0=0< t1, . . . , td−1< td=1). We derive the LDP for (l_tⁿ)t∈[0,1] from the LDP for the finite-dimensional marginals (l_tⁿ₀, . . . , l_tⁿ_d) in Section 3. This result is obtained using a projective limit approach taken from [4].

In Section 4 we prove the identity (7) so that (Lⁿ_t)t∈[0,1] is a mixture of LDS. Then we give the proof of Theorem 2, which is very close to the proof of Theorem 2.3 in [13]. Section 5 is devoted to applications of Theorem 2. We recover two classical examples of finite exchangeable random variables. We first consider the Curie–Weiss model, which is a well known toy model in statistical mechanics. Our analysis allows to consider both its microcanonical version (i.e., the uniform distribution on a set of allowed configurations), and its macrocanonical version (i.e., the classical Curie–Weiss model). These two aspects are connected via the principle of equivalence of ensembles.

The Curie–Weiss model is a paradigm for both exchangeable random variables and LD problems as can be seen, for example, in the fact that its internal fluctuations are studied by means of a de Finetti representation by Papangelou in [16], and by the same author using LD techniques in [17]. Another classical example is given by infinite exchangeable sequences, where Theorem 2 allows us to extend easily the result of [9].

We also show that the LDP’s for (Lⁿ_t)t∈[0,1] where X₁ⁿ, . . . , Xⁿ_n are respectively given by sampling with and without replacement have closely related rate functions. This completes, in a way, a result of Baxter and Jain [2]. Our last example concerns the random permutation of a discrete time stochastic process. An n-tuple (Y1, . . . , Yn) is transformed into(Xⁿ₁, . . . , Xⁿ_n)by the mechanism presented above, i.e.,X_iⁿ=Yσ (i)with σ =σⁿa random permutation on{1, . . . , n}, uniformly distributed and independent from (Y1, . . . , Yn). This appears to be a model for communication systems. A time-dependent signal Yⁿ is chopped into pieces of equal length (Y1, . . . , Yn) which are transmitted independently via different channels to the same destination. The signal is reconstructed according to the order of arrival into Xⁿ=(X₁ⁿ, . . . , Xⁿ_n), whose LD behavior is given by Theorem 2.

2. Large deviations for finite marginals of(l_tⁿ)t∈[0,1]

Let((y_iⁿ)1in)n∈N be a fixed triangular array of elements of and let d∈N^∗and t =(t0=0< t1, . . . , td−1< td =1). Our objective in this section is to prove that if µⁿ=¹_nⁿi=1δyⁿ_i

→w µthen(l_tⁿ₀, . . . , lⁿ_t_d)follows a LDP onM⁺()^d⁺¹, withl_tⁿ as in (4).

Fixing(l_tⁿ

0, . . . , l_tⁿ_d)is equivalent to choosing uniformly a partition of(y_iⁿ)1in among those withdclasses containing[ntj] − [ntj−1]elements, for 1jd. In other words, we must associate to everyy_iⁿ a valuej, under the strong condition that[ntj] − [ntj−1] items are associated to each j. First we relax the constraint on the cardinals of the d classes, and look for the LDP satisfied by the sequence of random measures

Lⁿ= 1 n

n i=1

δ(y_iⁿ,N_iⁿ), (9)

where the ((N_iⁿ)1in)n∈N are independent random variables defined on a probability space(Y,F, P ), with values in a Polish space0, identically distributed according to a

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lawλ. We will derive the LDP for(l_tⁿ₀, . . . , l_tⁿ_d)from the latter result by conditioning on the values ofN_iⁿ, thanks to a coupling.

LEMMA 1. – The distribution ofLⁿ underP obeys a LDP onM¹(×0)endowed with the topology of weak convergence, with good rate function

I1(ν, µ, λ)=

H (ν|µ⊗λ) ifν⁽¹⁾=µ,

∞ otherwise, (10)

whereν⁽¹⁾stands for the first marginal ofν.

Proof. – Let φ∈Cb(×0), where we denote by Cb(×0) the class of all real valued bounded continuous functions on×0. We have

logE

exp

n

×0

φ(u, v)Lⁿ(du×dv)

=logE

exp n

i=1

φy_iⁿ, N_iⁿ

= n

i=1

log

0

expφy_iⁿ, vλ(dv), then

8(φ)= lim

n→∞

1 nlogE

exp

n

×0

φ(u, v)Lⁿ(du×dv)

=

log

0

expφ(u, v)λ(dv)

µ(du) <∞.

Hence for all k∈Nall φ₁, . . . , φk ∈Cb(×0) and all λ₁, . . . , λk ∈R8(^k_i₌₁λiφi) is finite and differentiable in λ1, . . . , λk throughout R^k. Whence, according to part a) of Corollary 4.6.11 in [6], Lⁿ follows a LDP onX, the algebraic dual of Cb(×0), equipped with theCb(×0)-topology, with good rate function

8^∗(ν)= sup

φ∈Cb(×0)

φ, ν −8(φ),

where·,·stands as usual for

φ, ν =

×0

φdν. (11)

AsM¹(×0)is closed inX and8^∗(ν)= ∞onX\M¹(×0),Lⁿfollows a LDP on M¹(×0)equipped with the weak convergence topology, with good rate function8^∗. Let us identify8^∗. From Theorem A.5.4 in [10] we know that everyν∈M¹(×0) can be written as ν(du×dv)=ν⁽¹⁾(du)⊗ρ(u,dv), where ρ is a regular probability kernel.

First suppose thatν⁽¹⁾=µ. Then, there exists aφ∈Cb()such thatφ(u)ν⁽¹⁾(du)−

φ(u)µ(du)=1, so for every M >0 we define ψM ∈Cb(×0) by ψM(u, v)= Mφ(u)such that

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×0

ψM(u, v)ν(du×dv)−

log

0

expψM(u, v)λ(dv)

µ(du)

=M

φ(u)ν⁽¹⁾(du)−

φ(u)µ(du)

=M.

Whence we obtain in this case8^∗(ν)=I1(ν, µ, λ)= ∞by lettingM→ ∞.

Now suppose thatν⁽¹⁾=µ. By virtue of Jensen’s inequality, for anyφ∈Cb(×0) log

0

expφ(u, v)λ(dv)µ(du)

log

0

expφ(u, v)λ(dv)

µ(du).

Thus,

×0

φ(u, v)ν(du×dv)−log

0

expφ(u, v)λ(dv)µ(du)

×0

φ(u, v)ν(du×dv)−

log

0

expφ(u, v)λ(dv)

µ(du).

Then, according to the definition of H (· | ·), we obtain H (ν|µ⊗λ)8^∗(ν). So, if H (ν|µ⊗λ)= ∞, we necessarily have8^∗(ν)=I1(ν, µ, λ)= ∞. Otherwise, we can define

f (u, v)=d(µ⊗ρ) d(µ⊗λ) =dρ

dλµ⊗λ a.e.

For everyφ∈Cb(×0) Hρ(u,·)|λ

0

φ(u, v)ρ(u,dv)−log

0

expφ(u, v)λ(dv) µa.e., hence

Hρ(u,·)|λµ(du)

×0

φ(u, v)ν(du×dv)−

log

0

expφ(u, v)λ(dv)

µ(du),

soH (ρ(u,·)|λ)µ(du)8^∗(ν).

But, according to Fubini’s theorem

Hρ(u,·)|λµ(du)=

0

dρ dλlogdρ

dλdλ

dµ

=

×0

d(µ⊗ρ)

d(µ⊗λ)logd(µ⊗ρ)

d(µ⊗λ)d(µ⊗λ)

=H (ν|µ⊗λ).

soH (ν|µ⊗λ)8^∗(ν)and thenλ^∗(ν)=I1(ν, µ, λ). ✷

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We proceed now to the identification of each N_iⁿ (1in) with an element of a random partition indclasses of(y_iⁿ)1in. We suppose that0= {1, . . . , d}, that theN_iⁿ are distributed according toλ(j )=tj −tj−1=:<jt, and we define the continuous and injective map

F:M¹(×0) −→ M⁺()^d

ν(·,·) −→ ν·,{1}, ν·,{1,2}, . . . , ν·, 0.

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For everyn∈Nwe set

Sⁿ=F◦Lⁿ, (13) with Lⁿ as in (9). The vector of random measures Sⁿ is defined on (Y,F, P ) as in Lemma 1. An element ν=(νi)i∈0 ofM⁺()^d is said to be increasing when νi(A) νj(A)for allA∈Band all i, j∈0such thatij. For these elements ofM⁺()^dwe denote by<iνthe positive measureνi −νi−1, withν0=0.

COROLLARY 1. – The distribution ofSⁿunderP obeys a LDP onM⁺()^dequipped with the product topology of weak convergence, with good rate function

I2(ν, µ, t )=









 d

i=1

<iν()H

<iν

<iν() µ

+ d i=1

<iν()log<iν()

<it if

νis increasing, νd=µ,

∞ elsewhere.

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Proof. – Let M=F (M¹(×0))= {ν∈M⁺()^d, ν increasing andνd()=1}. Since F is continuous and injective, we deduce from Lemma 1 thatSⁿ follows a LDP on M⁺()^d endowed with the product topology of weak convergence, with good rate function

I¯2(ν, µ, t )=

I1(ν^∗, µ, λ) ifν∈Mandν=F (ν^∗),

∞ elsewhere,

whereI1is the rate function defined in (10).

Ifν /∈M, I¯2(ν, µ, t )=I2(ν, µ, t )= ∞. Let ν∈M. Then we have νd=ν^∗⁽¹⁾, the first marginal of ν^∗ and ifνd=µ, I2(ν, µ, t )= ¯I2(ν, µ, t )= ∞. Ifνd =µ thenν^∗ is absolutely continuous w.r.t.µ⊗λand

I¯₂(ν, µ, t )=I₁(ν^∗, µ, λ)

=H (ν^∗|µ⊗λ)

= d

i=1

ν^∗(dy, i)log ν^∗(dy, i) µ(dy)<it

= d

i=1

<iν()

<iν(dy)

<iν() log <iν(dy)/<iν() µ(dy)<it/<iν()

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= d

i=1

<iν()H

<iν

<iν() µ

+

d i=1

<iν()log<iν()

<it

=I2(ν, µ, t).

Hence we obtain the rate function of the LDP satisfied bySⁿ. ✷

Next we define a coupling procedure that allows us to derive from Sⁿ a random variable with the same law as (l_tⁿ

1, . . . , l_tⁿ

d). Let U_jⁿ be the number of j-valued N_iⁿ (j ∈ {1, . . . , d}), andTnbe the typical eventTn=^dj=1{U_jⁿ= [ntj] − [ntj−1]}. For every n∈Nwe define(N˜_iⁿ)1infrom(N_iⁿ)1in in the following way:

• IfU₁ⁿis greater than[nt1], we choose randomlyU₁ⁿ− [nt1]i’s among the ones with N_iⁿ=1, and we change the value 1 on thesei’s to the value 2.

• IfU₁ⁿis less than[nt1], we choose uniformly[nt1] −U₁ⁿindices among those such that N_iⁿ=2, and we change the associated N_iⁿ into 1. If there are not enough i’s such thatN_iⁿ=2, we choose the needed indices among those withN_iⁿ=3.

We call N¯_i,1ⁿ ∈ {1, . . . , d} the random variables resulting from this first step of the procedure. Now we define the random variables N¯_i,2ⁿ ∈ {1, . . . , d} resulting from the second step in the same way:

• If the number ofi’s withN¯_i,1ⁿ =2 is greater than[nt2] − [nt1], we choose uniformly the indices in excess, and we change the value 2 on thesei’s to the value 3.

• If the number of N¯_i,1ⁿ =2 is less than [nt₂] − [nt₁], we complete it by choosing uniformly indices among those such thatN¯_i,1ⁿ =3. If there are not enoughi’s such thatN_iⁿ=3, we choose the needed indices among those such thatN_iⁿ=4.

We carry on up to d−1, and we set N¯_i,jⁿ ∈ {1, . . . , d} for the ith random variable at thejth step of the coupling procedure. We put(N¯_i,0ⁿ )1in=(N_iⁿ)1inand we define the(N˜_iⁿ)1inbyN˜_iⁿ= ¯N_i,dⁿ ₋₁. For everyn∈Nwe note

L˜ⁿ= 1 n

n i=1

δ_(yn

i,N˜_iⁿ), (15)

and

Sⁿ=F◦ ˜Lⁿ, (16) withF as in (12).

LEMMA 2. – For everyn∈Nthe law ofSⁿ is the law ofSⁿ conditioned onTn, and for every measurableB⊂M⁺()^d we have

P Sⁿ∈B=Pⁿl_tⁿ₁, . . . , l_tⁿ_d∈B; µⁿ.

Proof. – Even if the random variables Sⁿ and (l_tⁿ₁, . . . , l_tⁿ_d) are defined on different probability spaces, their distribution onM⁺()^dhave the same finite supportAⁿ, and it is also the support of the distribution ofSⁿconditioned onTn. Since(x₁ⁿ, . . . , x_nⁿ)results from a sampling without replacement all possible(lⁿ_t

1, . . . , l_tⁿ_d)are equally-likely, thus for

(11)

everyρ∈Aⁿ

Pⁿl_tⁿ

1, . . . , l_tⁿ_d=ρ;µⁿ= 1

|Aⁿ|. The cardinal ofAⁿ might not be

d i=1

n− [nti−1] [nti] − [nti−1]

because of possible ties among(yⁿ₁, . . . , y_nⁿ). In the same time, as the law of(N₁ⁿ, . . . , N_nⁿ) conditioned onTnis uniform on its support, for everyρ∈Aⁿ

PSⁿ=ρ|Tn

= 1

|Aⁿ|.

Hence it is then sufficient to prove thatSⁿis uniformly distributed onAⁿ. For allρ,γ ∈ Im(Sⁿ)there areu=(ui)1in andv=(vi)1insuch that we have{Sⁿ=ρ} = { ˜N₁ⁿ= u1, . . . ,N˜_nⁿ=un}and{Sⁿ=γ} = { ˜N₁ⁿ=v1, . . . ,N˜_dⁿ=vn}, and there is a permutationσ on{1, . . . , n}such that for alli ui =vσ (i). Hence,P (Sⁿ=ρ)=P (Sⁿ=γ )if and only if (N˜_iⁿ)1in isn-exchangeable. In order to prove it we introduce the following notations:

• Vu^v(j )stands for the event:

“Thejth step of the coupling procedure changes(N¯_i,jⁿ ₋₁)1in=uto(N¯_i,jⁿ )1in

=v”.

• For all 1inand for all 1qdwe callY_i^q=(N¯_i,0ⁿ , . . . ,N¯_i,qⁿ ₋₁)∈ {1, . . . , d}^q the random vector that records the values associated toiduring the procedure.

Note that what matters inVu^v(j )is the number ofk-valuedui’s andvi’s inuand vfor eachk∈ {j, . . . , d}, not the value of eachui andvi. Hence, for every permutationσ on {1, . . . , n}we haveP (Vu^v(j ))=P (V_{σ (u)}^{σ (v)}(j )), whereσ (u)=(uσ (1), . . . , uσ (n)).

We prove by induction on q that for every 1 q d, (Y_i^q)1in is n-exchange- able. For q =1, the (N¯_i,0ⁿ )1in are independent and identically distributed, whence (Y_i¹)1inisn-exchangeable. Suppose the property holds for a fixedq (1qd−1):

(Y_i^q=(N¯_i,0ⁿ , . . . ,N¯_i,qⁿ ₋₁))1inisn-exchangeable. Let(u^j_i)∈Mn,q+1(0), we denote by u_i itsith row and byu^j itsjth column. For every permutationσ on{1, . . . , n}

PY_i^q⁺¹=u_i, 1in

=PN¯_i,0ⁿ =u⁰_i, . . . ,N¯_i,qⁿ =u^q_i,1in

=PN¯_i,0ⁿ =u⁰_i, . . . ,N¯_i,qⁿ ₋₁=u^q_i⁻¹, V_u^u_q−1^q (q), 1in

=PV_u^uq^q−1(q)| ¯N_i,qⁿ ₋₁=u^q_i⁻¹, 1inPN¯_i,0ⁿ =u⁰_i, . . . ,N¯_i,qⁿ ₋₁=u^q_i⁻¹, 1in

=PV^{σ (u}

q−1)

σ (u^q⁻¹)(q)PN¯_{σ (i),0}ⁿ =u⁰_i, . . . ,N¯_{σ (i),q}ⁿ ₋₁=u^q_i⁻¹, 1in

=PY_{σ (i)}^q⁺¹=u_i,1in.