# Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution

Authors: Richard S. Ellis; Shlomo Ta’asan
Journal: International Journal of Stochastic Analysis
 (2015)

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## Abstract

In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Givenb∈N and c>b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice, where K/N equals c. We focus on configurations for which each site is occupied by a minimum of b particles. The main result is the large deviation principle (LDP), in the limit K→∞ and N→∞ with K/N=c, for a sequence of random, number-density measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy R(θ∣ρ∗), where θ is a possible asymptotic configuration of the number-density measures and ρ∗ is a Poisson distribution with mean c, restricted to the set of positive integers n satisfying n≥b. This LDP implies that ρ∗ is the equilibrium distribution of the number-density measures, which in turn implies that ρ∗ is the equilibrium distribution of the random variables that count the droplet sizes.


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## Body

## 1. Introduction

This paper is motivated by a natural question for a basic model of a droplet. Givenb∈N and c>b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice ΛN={1,2,…,N}. Under the assumption that K/N=c and that each site is occupied by a minimum of b particles, what is the equilibrium distribution, as N→∞, of the number of particles per site? We prove in Corollary 3 that this equilibrium distribution is a Poisson distribution, with mean c, restricted to the set of positive integers n satisfying n≥b. As we explain near the end of the Introduction, this equilibrium distribution has important applications to technologies using sprays and powders.As in many other models in statistical mechanics, we can identify the equilibrium distribution by exhibiting it as the unique minimum point of a rate function in a large deviation principle (LDP). Other models for which this procedure can be implemented are discussed at the end of the Introduction.For the droplet model we prove the LDP for a sequence of random probability measures, called number-density measures, which are the empirical measures of a sequence of dependent random variables that count the droplet sizes. This LDP is stated in Theorem1. Our proof is self-contained and starts from first principles, using techniques that are familiar in applied mathematics and statistical mechanics. For example, the proof of the local large deviation estimate in Theorem 5, a key step in the proof of the LDP for the number-density measures, is based on combinatorics, Stirling’s formula, and Laplace asymptotics.Our use of combinatorial methods goes back to Boltzmann in his work on the discrete ideal gas. He calculated the Maxwell-Boltzmann equilibrium distribution for this system by analyzing the asymptotic behavior of a particular multinomial coefficient [1]. Starting with Boltzmann’s work, combinatorial methods have remained an important tool in both statistical mechanics and in the theory of large deviations, offering insights into a wide variety of physical and mathematical phenomena via techniques that are elegant, powerful, and often elementary. In applications to statistical mechanics, this state of affairs is explained by the observation that “many fundamental questions … are inherently combinatorial, … including the Ising model, the Potts model, monomer-dimer systems, self-avoiding walks and percolation theory” [2]. For the two-dimensional Ising model and other exactly soluble models, [3, 4] are recommended.A similar situation holds in the theory of large deviations. For example, Section2.1 of [5] discusses combinatorial techniques for finite alphabets and points out that because of the concreteness of these applications the LDPs are proved under much weaker conditions than the corresponding results in the general theory, into which the finite-alphabet results give considerable insight. The text [6] devotes several early sections to large deviation results for i.i.d. random variables having a finite state space and proved by combinatorial methods, including a sophisticated, level-3 result for the empirical pair measure.In order to formulate the LDP for the number-density measures in our droplet model, a standard probabilistic model is introduced. The configuration space is the setΩN=ΛNK consisting of all ω=(ω1,ω2,…,ωK), where ωi denotes the site in ΛN occupied by the ith particle. The cardinality of ΩN equals NK. Denote by PN the uniform probability measure that assigns equal probability 1/NK to each of the NK configurations ω∈ΩN. The asymptotic analysis of the droplet model involves the two random variables, which are functions of the configuration ω∈ΩN: for l∈ΛN, Kl(ω) denotes the number of particles occupying the site l in the configuration ω; for j∈N∪{0}, Nj(ω) denotes the number of sites l∈ΛN for which Kl(ω)=j.We focus on the subset ofΩN consisting of all configurations ω for which every site of ΛN is occupied by at least b particles. Because of this restriction Nj(ω) is indexed by j∈Nb={n∈Z:n≥b}. It is useful to think of each particle as having one unit of mass and of the set of particles at each site l as defining a droplet. With this interpretation, for each configuration ω, Kl(ω) denotes the mass or size of the droplet at site l. The jth droplet class has Nj(ω) droplets and mass jNj(ω). Because the number of sites in ΛN equals N and the sum of the masses of all the droplet classes equals K, the following conservation laws hold for such configurations: (1)∑j∈NbNjω=N,∑j∈NbjNjω=K.In addition, since the total number of particles is K, it follows that ∑l∈ΛNKl=K. These equality constraints show that the random variables Nj and Kl are not independent.In order to carry out the asymptotic analysis of the droplet model, we introduce a quantitym=m(N) that converges to ∞ sufficiently slowly with respect to N; specifically, we require that m(N)2/N→0 as N→∞. In terms of b and m we define the subset ΩN,b,m of ΩN consisting of all configurations ω for which every site of ΛN is occupied by at least b particles and at most m of the quantities Nj(ω) are positive. This second condition is a useful technical device that allows us to control the errors in several estimates. In Appendix D of [7] we present evidence supporting the conjecture that this condition can be eliminated. The discussion in that appendix involves a number of interesting topics including Stirling numbers of the second kind (see [8, pp. 96-97] and [9, §5.4]) and their asymptotic behavior [10, Example 5.4].The random quantities in the droplet model for which we formulate an LDP are the number-density measuresΘN,b. For ω∈ΩN,b,m these random probability measures assign to j∈Nb the probability Nj(ω)/N, which is the number density of the jth droplet class. Because of the two conservation laws in (1) and because K/N=c, for ω∈ΩN,b,m, ΘN,b(ω) is a probability measure on Nb={n∈Z:n≥b} having mean c. Thus ΘN,b takes values in PNb,c, which is defined to be the set of probability measures on Nb having mean c.The probability measurePN,b,m defining the droplet model is obtained by restricting the uniform measure PN to the set of configurations ΩN,b,m. Thus PN,b,m equals the conditional probability PN(·∣ΩN,b,m). In the language of statistical mechanics PN,b,m defines a microcanonical ensemble that incorporates the conservation laws for number and mass expressed in (1).A natural question is to determine two equilibrium distributions: the equilibrium distributionρ∗ of the number-density measures and the equilibrium distribution ρ∗∗=∑j∈Nbρj∗∗δj of the droplet-size random variables Kl. These distributions are defined by the following two limits: for any ε>0, any l∈ΛN, and all j∈Nb(2)limN→∞⁡PN,b,mΘN,b∈Bρ∗,ε⟶1,limN→∞⁡PN,b,mKl=j=ρj∗∗, where B(ρ∗,ε) denotes the open ball with center ρ∗ and radius ε defined with respect to an appropriate metric on PNb,c. As we prove, the equilibrium distributions of ΘN,b and Kl coincide. As in many models in statistical mechanics, an efficient way to determine the equilibrium distribution ΘN,b is to prove an LDP for ΘN,b, which we carry out in Theorem 1. This theorem is the main result in the paper.The content of Theorem1 is the following: as N→∞, the sequence of number-density measures ΘN,b satisfies the LDP on PNb,c with respect to the measures PN,b,m. The rate function is the relative entropy R(θ∣ρb,α) of θ∈PNb,c with respect to the Poisson distribution ρb,α on Nb having components ρb,α;j=[Zb(α)]-1·αj/j! for j∈Nb. In this formula Zb(α) is the normalization that makes ρb,α a probability measure, and α equals the unique value αb(c) for which ρb,αb(c) has mean c [Theorem A.2]. Using the fact that R(θ∣ρb,αb(c)) equals 0 at the unique measure θ=ρb,αb(c), we apply the LDP for ΘN,b to conclude in Theorem 2 that ρb,αb(c) is the equilibrium distribution of ΘN,b. Corollary 3 then implies that ρb,αb(c) is also the equilibrium distribution of Kl.The spacePNb,c is the most natural space on which to formulate the LDP for ΘN,b in Theorem 1. Not only is PNb,c the smallest convex set of probability measures containing the range of ΘN,b for all N∈N, but also the union over N∈N of the range of ΘN,b is dense in PNb,c. As we explain in part (a) of Theorem 4, PNb,c is not a complete, separable metric space, a situation that prevents us from directly applying general results in the theory of large deviations that require the setting of a complete, separable metric space.The droplet model is defined in Section2. Step 1 in the proof of the LDP for ΘN,b is to derive the local large deviation estimate in part (b) of Theorem 5. This local estimate, one of the centerpieces of the paper, gives information not available in the LDP for ΘN,b, which involves global estimates. Step 2 is to lift the local large deviation estimate to the large deviation limit for ΘN,b lying in open balls and certain other subsets of PNb,c while Step 3 is to lift the large deviation limit for open balls and certain other subsets to the LDP for ΘN,b stated in Theorem 1. Steps 2 and 3 are explained in Section 4.Details of Steps2 and 3 as well as other routine proofs are omitted from the present paper. They appear in the unpublished companion paper [7], which also contains additional background material. The paper [1] explores how our work on the droplet model was inspired by the work of Ludwig Boltzmann on a simple model of a discrete ideal gas. The main connection is via the local large deviation estimate in part (b) of Theorem 5. When b=0, the LDP for a path version of Θn,0 with K=tN and t>0 varying appears in [11, 12].The main application of the results in this paper is to technologies using sprays and powders, which are ubiquitous in many fields, including agriculture, the chemical and pharmaceutical industries, consumer products, electronics, manufacturing, material science, medicine, mining, paper making, the steel industry, and waste treatment. In this paper we focus on sprays; our theory also applies to powders with only changes in terminology [13]. The behavior of sprays might be complex depending on various parameters including evaporation, temperature, and viscosity. Our goal here is to consider the simplest model where the only assumption is made on the average size of droplets in the spray. In many situations it is important to have good control over the sizes of the droplets, which can be translated into properties of probability distributions. The size distributions are important because they determine reliability and safety in each particular application.Interestingly, there does not seem to be a rigorous theory that predicts the equilibrium distribution of droplet sizes, analogous to the Maxwell-Boltzmann distribution of energy levels in a discrete ideal gas [14, 15]. Our goal in the present paper is to provide such a theory. We do so by focusing on one aspect of the problem related to the relative entropy, an approach that characterizes the equilibrium distribution of droplet sizes as being a Poisson distribution restricted to Nb. We expect that this distribution will be important in experimental observations. A full understanding of droplet behavior under dynamic conditions requires treating many other aspects and is beyond the scope of this paper. We plan to apply the ideas in this paper to understand the entropy of dislocation networks [16].The importance of predicting droplet size can be seen from the wide range of applications utilizing sprays [17, 18]. Because of the importance of this problem, novel approaches for measuring size distribution of droplet size in sprays have been developed [19–23]. What makes the problem of predicting droplet size particularly interesting is the complexity of droplet-size distribution, which is attributed to many factors such as temperature and viscosity. As [24] shows, even the nozzle plays a significant role in the outcome. Many theoretical tools used to understand the distribution of droplet size in sprays include entropy [25], which also plays a key role in the present paper.We end the Introduction by expanding on a comment made at the beginning of this section. This comment concerns one of the main applications of large deviation theory in statistical mechanics, which is to identify the equilibrium distribution or distributions of a model as the minimum point(s) of the rate function in an LDP for the model. This procedure is also useful to study phase transitions in the model, which concern how the structure of the set of equilibrium distributions changes as the parameters defining the model change. There are numerous other models for which this procedure has been used. They include the following three lattice spin models: the Curie-Weiss spin system, the Curie-Weiss-Potts model, and the mean-field Blume-Capel model, which is also known as the mean-field BEG model. As explained in the respective Sections6.6.1, 6.6.2, and 6.6.3 of [26], the large deviation analysis shows that each of these three models has a different phase transition structure. Details of the analysis for the three models are given in the references [6, §IV.4], [27–29]. Section 9 of [30] outlines how large deviation theory can be applied to determine equilibrium structures in statistical models of two-dimensional turbulence. Details of this analysis are given in [31].


## 2. Definition of Droplet Model and Main Theorem

After defining the droplet model, we state the main theorem in the paper, Theorem1. The content of this theorem is the LDP for the sequence of random, number-density measures, which are the empirical measures of a sequence of dependent random variables that count the droplet sizes in the model. As we show in Theorem 2 and in Corollary 3, the LDP enables us to identify a Poisson distribution as the equilibrium distribution both of the number-density measures and of the droplet-size random variables. In Theorem 4 we prove a number of properties of two spaces of probability measures in terms of which the LDP for the number-density measures is formulated.We start by fixing parametersb∈N∪{0} and c∈(b,∞). The droplet model is defined by a probability measure PN,b parameterized by N∈N and the nonnegative integer b. The measure depends on two other positive integers, K and m, where 2≤m≤N<K. Both K and m are functions of N in the large deviation limit N→∞. In this limit we take K→∞ and N→∞, where K/N, the average number of particles per site, equals c. Thus K=Nc. In addition, we take m→∞ sufficiently slowly by choosing m to be a function m(N) satisfying m(N)→∞ and m(N)2/N→0 as N→∞; for example, m(N)=Nδ for some δ∈(0,1/2). Throughout this paper we fix such a function m(N). The parameter b and the function m=m(N) first appear in the definition of the set of configurations ΩN,b,m in (3), where these quantities will be explained.BecauseK and N are integers, c must be a rational number. This in turn imposes a restriction on the values of N and K. If c is a positive integer, then N→∞ along the positive integers and K→∞ along the subsequence K=cN. If c=x/y, where x and y are relatively prime, positive integers with y≥2, then N→∞ along the subsequence N=yn for n∈N and K→∞ along the subsequence K=cN=xn. Throughout this paper, when we write N∈N or N→∞, it is understood that N and K satisfy the restrictions discussed here.In the droplet modelK distinguishable particles are placed, each with equal probability 1/N, onto the sites of the lattice ΛN={1,2,…,N}. This simple description corresponds to a simple probabilistic model. The configuration space is the set ΩN=ΛNK consisting of all sequences ω=(ω1,ω2,…,ωK), where ωi∈ΛN denotes the site in ΛN occupied by the ith particle. Let ρ(N) be the measure on ΛN that assigns equal probability 1/N to each site in ΛN, and let PN=(ρ(N))K be the product measure on ΩN with equal one-dimensional marginals ρ(N). Thus PN is the uniform probability measure that assigns equal probability 1/NK to each of the NK configurations ω∈ΩN; for subsets A of ΩN we have PN(A)=card⁡(A)/NK, where card denotes cardinality.The asymptotic analysis of the droplet model involves two random variables. Forl∈ΛN and ω∈ΩN, Kl(ω) denotes the number of particles occupying site l in the configuration ω. For j∈N∪{0} and ω∈ΩN, Nj(ω) denotes the number of sites l∈ΛN for which Kl(ω)=j. The dependence of Kl(ω) and Nj(ω) on N is not indicated in the notation. Because the distributions of both random variables depend on N, both Kl and Nj form triangular arrays.We now specify the role played by the nonnegative integerb, first focusing on the case where b is a positive integer. The case where b=0 is discussed later. For ω∈ΩN, in general there exist sites l∈ΛN for which Kl(ω)=0; that is, sites that are occupied by 0 particles. The next step in the definition of the droplet model is to restrict to a subset ΩN,b,m of configurations ω∈ΩN for which every site is occupied by at least b particles and the following constraint holds: for any configuration ω∈ΩN,b,m at most m of the components Nj(ω) are positive, where m=m(N)→∞ and m(N)2/N→0 as N→∞. Because for ω∈ΩN,b,m every site l∈ΛN is occupied by at least b particles, we have Kl(ω)≥b and Nj(ω) is indexed by j∈Nb={n∈Z:n≥b}. We denote by N(ω) the sequence {Nj(ω),j∈Nb} and define Nω+=card{j∈Nb:Nj(ω)≥1}. In terms of this notation (3)ΩN,b,m=ω∈ΩN:Klω≥b,∀l∈ΛN,Nω+≤m=mN.The constraint restricting the number of positive components ofN(ω) is a useful technical device that allows us to control the errors in several estimates. In Appendix D of [7] we give evidence supporting the conjecture that this restriction can be eliminated.Whenb is a positive integer, for each ω∈ΩN,b,m, each site in ΛN is occupied by at least b particles. In this case it is useful to think of each particle as having one unit of mass and of the set of particles at each site l as defining a droplet. With this interpretation, for each configuration ω, Kl(ω) denotes the mass or the size of the droplet at site l. The jth droplet class has Nj(ω) droplets and mass jNj(ω). Because the number of sites in ΛN equals N and the sum of the masses of all the droplet classes equals K, it follows that the quantities Nj(ω) satisfy the two conservation laws in (1) for all ω∈ΩN,b,m.We now consider the modifications that must be made in these definitions whenb=0. In this case the first constraint in the definition of ΩN,b,m disappears because we allow sites to be occupied by 0 particles, and therefore Nj(ω) is indexed by j∈N0=N∪{0}. On the other hand, we retain the second constraint in the definition of ΩN,0,m, which requires that for any configuration ω∈ΩN,0,m at most m of the components Nj(ω) for j∈N0 are positive. When b=0, the definition of ΩN,0,m becomes ΩN,0,m={ω∈ΩN:Nω+≤m=m(N)}. Because the choice b=0 allows sites to be empty, we lose the interpretation of the set of particles at each site as being a droplet. However, for ω∈ΩN,0,m the two conservation laws in (1) continue to hold.For the remainder of this paper we work with any fixed nonnegative integerb. The probability measure PN,b,m defining the droplet model is obtained by restricting the uniform measure PN to the set ΩN,b,m. Thus PN,b,m equals the conditional probability PN(·∣ΩN,b,m). For subsets A of ΩN,b,m, PN,b,m(A) takes the form (4)PN,b,mA=PNA∣ΩN,b,m=1card⁡ΩN,b,m·card⁡A.Having defined the droplet model, we introduce the random probability measures whose large deviations we will study. Forω∈ΩN,b,m these measures are the number-density measures ΘN,b that assign to j∈Nb the probability Nj(ω)/N. This ratio represents the number density of droplet class j. Thus for any subset A of Nb(5)ΘN,bω,A=∑j∈AΘN,b;jω,whereΘN,b;jω=NjωN.By the two formulas in (1) ∑j∈NbΘN,b;j(ω)=1 and ∑j∈NbjΘN,b;j(ω)=K/N=c. Thus ΘN,b(ω) is a probability measure on Nb having mean c.We next introduce several spaces of probability measures that arise in the large deviation analysis of the droplet model.PNb denotes the set of probability measures on Nb={n∈Z:n≥b}. Thus θ∈PNb has the form ∑j∈Nbθjδj, where the components θj satisfy θj≥0 and ∑j∈Nbθj=1. We say that a sequence of measures {θ(n),n∈N} in PNb converges weakly to θ∈PNb, and write θ(n)⇒θ, if, for any bounded function f mapping Nb into R, ∫Nbfdθ(n)→∫Nbfdθ as n→∞. PNb is topologized by the topology of weak convergence. There is a standard technique for introducing a metric structure on PNb for which we quote the main facts. Because Nb is a complete, separable metric space with metric d(x,y)=x-y, there exists a metric π on PNb called the Prohorov metric with the following two properties: (1) convergence with respect to the Prohorov metric is equivalent to weak convergence [32, Thm. 3.3.1]; (2) with respect to the Prohorov metric, PNb is a complete, separable metric space [32, Thm. 3.1.7].We denote byPNb,c the set of measures in PNb having mean c. Thus θ∈PNb,c has the form ∑j∈Nbθjδj, where the components θj satisfy θj≥0, ∑j∈Nbθj=1, and ∑j∈Nbjθj=c. The number-density measures ΘN,b defined in (5) take values in PNb,c.According to part (a) of Theorem4, PNb,c is not a closed subset of PNb. Hence it is natural to introduce the closure of PNb,c in PNb. As we prove in part (b) of Theorem 4, the closure of PNb,c in PNb equals PNb,[b,c], which is the set of measures in PNb having mean lying in the closed interval [b,c]. Being the closure of the relatively compact, separable metric space PNb,c, PNb,[b,c] is a compact, separable metric space with respect to the Prohorov metric. This space appears in the formulation of the large deviation upper bound in part (c) of Theorem 1.We next state Theorem1, which is the LDP for the sequence of distributions PN,b,m(ΘN,b∈dθ) on PNb,c as N→∞. The rate function in the LDP is the relative entropy of θ with respect to the Poisson distribution ρb,αb(c)=∑j∈Nb‍ρb,αb(c);jδj defined in (7), where each ρb,αb(c);j>0. Thus any θ∈PNb,c is absolutely continuous with respect to ρb,αb(c). For θ∈PNb,c the relative entropy of θ with respect to ρb,αb(c) is defined by (6)Rθ∣ρb,αbc=∑j∈Nbθjlog⁡θjρb,αbc;j. If θj=0, then θjlog⁡(θj/ρb,αb(c);j)=0. For j∈Nb the components of the measure ρb,αb(c) appearing in the LDP have the form (7)ρb,αbc;j=1Zbαbc·αbcjj!, where αb(c)∈(0,∞) is chosen so that ρb,αb(c) has mean c and Zb(αb(c)) is the normalization making ρb,αb(c) a probability measure; thus Z0(α0(c))=eα0(c) and, for b∈N, Zb(αb(c))=eαb(c)-∑j=0b-1[αb(c)]j/j!. As we show in Theorem A.2, there exists a unique value of αb(c).As a consequence of the fact thatPNb,c is not closed in PNb, the large deviation upper bound takes two forms depending on whether the subset F of PNb,c is compact or whether F is closed. When F is compact, in part (b) we obtain the standard large deviation upper bound for F. When F is closed, in part (c) we obtain a variation of the standard large deviation upper bound, which, when F is compact, coincides with the upper bound in part (b). The refinement in part (c) is important. It is applied in the proof of Theorem 2 to show that ρb,αb(c) is the equilibrium distribution of the number-density measures ΘN,b. In turn, Theorem 2 is applied in the proof of Corollary 3 to show that ρb,αb(c) is the equilibrium distribution of the droplet-size random variables Kl.In the next theorem we assume thatm is the function m(N) appearing in the definition of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. The assumption that m(N)2/N→0 is used to control error terms in Lemmas 6 and 7 in the present paper and in Lemma B.3 in [7]. This assumption on m(N) is optimal in the sense that it is a minimal assumption guaranteeing that error terms in parts (a) and (b) of Lemma B.3 in [7] converge to 0. In the next theorem, for A a subset of PNb,c or PNb,[b,c] we denote by R(A∣ρb,αb(c)) the infimum of R(θ∣ρb,αb(c)) over θ∈A.Theorem 1.
Fix a nonnegative integerb and a rational number c∈(b,∞). Let m be the function m(N) appearing in the definition of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. Let ρb,αb(c)∈PNb,c be the distribution having the components defined in (7). Then as N→∞, with respect to the measures PN,b,m, the sequence ΘN,b satisfies the LDP on PNb,c with rate function R(θ∣ρb,αb(c)) in the following sense.(a)
R(θ∣ρb,αb(c)) maps PNb,c into [0,∞] and has compact level sets in PNb,c; that is, for any M<∞ the set {θ∈PNb,c:R(θ∣ρb,αb(c))≤M} is compact.(b)
For any compact subsetF of PNb,c we have the large deviation upper bound(8)limsupN→∞⁡1Nlog⁡PN,b,mΘN,b∈F≤-RF∣ρb,αbc.(c)
For any closed subsetF of PNb,c, let F¯ denote the closure of F in PNb,[b,c]. We have the large deviation upper bound(9)limsupN→∞⁡1Nlog⁡PN,b,mΘN,b∈F≤-RF¯∣ρb,αbc.(d)
For any open subsetG of PNb,c we have the large deviation lower bound(10)liminfN→∞⁡1Nlog⁡PN,b,mΘN,b∈G≥-RG∣ρb,αbc.The properties ofR(θ∣ρb,αb(c)) in part (a) are proved in [33, Lem. 1.4.1] and part (a) of Theorem A.1. The basic step in proving the large deviation bounds in parts (b)–(d) is the local large deviation estimate in part (b) of Theorem 5. As explained in Section 4, this local estimate is lifted to large deviation limits involving open balls stated in Theorem 8, which in turn are used to derive the bounds in parts (b)–(d) of Theorem 1.In the next theorem we use the large deviation upper bound in part (c) of Theorem1 to prove that the Poisson distribution ρb,αb(c) is the equilibrium distribution of the number-density measures ΘN,b. In this theorem [Bπ(ρb,αb(c),ε)]c denotes the complement in PNb,c of the open ball Bπ(ρb,αb(c),ε)={ν∈PNb,c:π(ρb,αb(c),ν)<ε}. [B^π(ρb,αb(c),ε)]c denotes the complement in PNb,[b,c] of the open ball B^π(ρb,αb(c),ε)={ν∈PNb,[b,c]:π(ρb,αb(c),ν)<ε}.Theorem 2.
One assumes the hypotheses of Theorem1. The following results hold for any ε>0.(a)
The quantityx∗=inf{R(θ∣ρb,αb(c)):θ∈[B^π(ρb,αb(c),ε)]c} is strictly positive.(b)
For any numbery in the interval (0,x∗) and all sufficiently large N(11)PN,b,mΘN,b∈Bπρb,αbc,εc≤exp⁡-Ny⟶0asN⟶∞.This upper bound implies that, as N→∞, PN,b,m(ΘN,b∈Bπ(ρb,αb(c),ε))→1 and for any bounded, continuous function g mapping PNb,c into R(12)limN→∞⁡∫ΩN,b,mgΘN,bdPN,b,m=gρb,αbc. These two limits allow us to interpret the Poisson distribution ρb,αb(c) as the equilibrium distribution of the number-density measures ΘN,b with respect to PN,b,m.Proof.
The starting point is the large deviation upper bound in part (c) of Theorem1 applied to the closed set [Bπ(ρb,αb(c),ε)]c, which is a subset of [B^π(ρb,αb(c),ε)]c. We denote the closure of [Bπ(ρb,αb(c),ε)]c in PNb,[b,c] by [Bπ(ρb,αb(c),ε]c¯. Since [Bπ(ρb,αb(c),ε)]c¯⊂[B^π(ρb,αb(c),ε)]c, the large deviation upper bound in part (c) of Theorem 1 takes the form (13)limsupN→∞⁡1Nlog⁡PN,b,mΘN,b∈Bπρb,αbc,εc≤-RBπρb,αbc,εc¯∣ρb,αbc≤-RB^πρb,αbc,εc∣ρb,αbc. We now prove part (a) of Theorem 2. Since R(θ∣ρb,αb(c)) is lower semicontinuous on PNb,[b,c] and has compact level sets in PNb,[b,c] [33, Lem. 1.4.3(b)–(c)], it attains its infimum x∗ on the closed set [B^π(ρb,αb(c),ε)]c. If x∗=0, then there would exist θ∈[B^π(ρb,αb(c),ε)]c such that R(θ∣ρb,αb(c))=0. But on PNb,[b,c], R(θ∣ρb,αb(c)) attains its infimum of 0 at the unique measure θ=ρb,αb(c) [33, Lem. 1.4.1]. This contradicts the fact that ρb,αb(c)∉[B^π(ρb,αb(c),ε)]c, completing the proof of part (a). The inequality in part (b) is an immediate consequence of part (a) and the large deviation upper bound (13). This inequality yields the limit PN,b,m(ΘN,b∈Bπ(ρb,αb(c),ε))→1, which in turn implies (12). The proof of Theorem 2 is complete.We now apply Theorem2 to prove that ρb,αb(c) is also the equilibrium distribution of the random variables Kl, which count the droplet sizes at the sites of ΛN. This is the content of the next corollary. A fact needed in the proof is that ΘN,b is the empirical measure of these random variables; that is, for ω∈ΩN,b,m, ΘN,b(ω) assigns to subsets A of Nb the probability ΘN,b(ω,A)=N-1∑l=1NδKl(ω)(A). This representation is valid because both ΘN,b(ω) and the empirical measure assign to j∈ΛN the probability Nj(ω)/N.Corollary 3.
One assumes the hypotheses of Theorem1. Then for any site l∈ΛN and any j∈Nb(14)limN→∞⁡PN,b,mKl=j=ρb,αbc;j=1Zbαbc·αbcjj!.Proof.
Since the random variablesKl are identically distributed, it suffices to prove the corollary for l=1. For fixed j∈Nb, the limit (12) with g(θ)=θj yields (15)limN→∞⁡PN,b,mK1=j=limN→∞⁡1N∑l=1N∫ΩN,b,m1jKldPN,b,m=limN→∞⁡∫ΩN,b,mΘN,b;jdPN,b,m=ρb,αbc;j. This completes the proof.The last theorem in this section proves several properties ofPNb,c and PNb,[b,c] with respect to the Prohorov metric that are needed in the paper.Theorem 4.
Fix a nonnegative integerb and a real number c∈(b,∞). The metric spaces PNb,c and PNb,[b,c] have the following properties.(a)
PNb,c, the set of probability measures on Nb having mean c, is a relatively compact, separable subset of PNb. However, PNb,c is not a closed subset of PNb and thus is not a compact subset or a complete metric space.(b)
PNb,[b,c], the set of probability measures on Nb having mean lying in the closed interval [b,c], is the closure of PNb,c in PNb. PNb,[b,c] is a compact, separable subset of PNb.Proof.
(a) Forξ∈N satisfying ξ≥b let Ψξ denote the compact subset {b,b+1,…,ξ} of Nb, and let [Ψξ]c denote its complement. For any θ∈PNb,c(16)c=∑j∈Nbjθj≥∑j≥ξ+1jθj≥ξ∑j≥ξ+1θj=ξθΨξc. It follows that PNb,c is tight; that is, for any ε>0 there exists ξ∈N such that θ([Ψξ]c)<ε for all θ∈PNb,c. Prohorov’s theorem implies that PNb,c is relatively compact [32, Thm. 3.2.2]. The separability of PNb,c is proved in Corollary B.2 in [7].
We now prove thatPNb,c is not a closed subset of PNb by exhibiting a sequence θ(n)∈PNb,c having a weak limit that does not lie in PNb,c. Let θ be any measure in PNb with mean β∈[b,c); thus θ∉PNb,c. The sequence (17)θn=n-cn-βθ+c-βn-βδnforn∈N,n>c has the property that θ(n)∈PNb,c and that θ(n)⇒θ∉PNb,c. This completes the proof of part (a).
(b) SincePNb,c is a separable subset of PNb and PNb,c is dense in PNb,[b,c], it follows that PNb,[b,c] is separable. We prove that PNb,[b,c] is the closure of PNb,c in PNb. Let θ(n) be a sequence in PNb,c converging weakly to θ∈PNb. Since θ(n)⇒θ implies that θj(n)→θj for each j∈Nb, Fatou’s lemma implies that c=liminfn→∞⁡〈θ(n)〉≥〈θ〉, where 〈θ(n)〉 and 〈θ〉 denote the means of θ(n) and θ. Since for any θ∈PNb we have 〈θ〉≥b, it follows that c≥〈θ〉≥b. This shows that the closure of PNb,c in PNb is a subset of PNb,[b,c]. We next prove that PNb,[b,c] is a subset of the closure of PNb,c in PNb by showing that for any θ∈PNb,[b,c] there exists a sequence θ(n)∈PNb,c such that θ(n)⇒θ. If 〈θ〉=c, then we choose θ(n)=θ for all n∈N. If 〈θ〉=β∈[b,c), then we use the sequence θ(n) in (17), which converges weakly to θ. We conclude that θ lies in the closure of PNb,c and thus that PNb,[b,c] is a subset of the closure of PNb,c in PNb. This completes the proof of part (b). The proof of Theorem 4 is done.In the next section we present the local large deviation estimate that will be used in Section4 to prove the LDP for ΘN,b in Theorem 1.


## 3. Local Large Deviation Estimate Yielding Theorem1

The main result needed to prove the LDP in Theorem1 is the local large deviation estimate stated in part (b) of Theorem 5. The first step is to introduce a set AN,b,m that plays a central role in this paper. Fix a nonnegative integer b and a rational number c∈(b,∞). Given N∈N define K=Nc and let m be the function appearing in the definition of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. Define Nb={n∈Z:n≥b}; thus N0 is the set of nonnegative integers. Let ν be a sequence {νj,j∈Nb} for which each νj∈N0; thus ν∈N0Nb. We define AN,b,m to be the set of ν∈N0Nb satisfying (18)∑j∈Nbνj=N,∑j∈Nbjνj=K,ν+≤m=mN, where ν+=card⁡{j∈Nb:νj≥1}. Because νj∈N0, the two sums involve only finitely many terms.Forω∈ΩN,b,m the components ΘN,b;j(ω) of the number-density measure defined in (5) are Nj(ω)/N for j∈Nb, where Nj(ω) denotes the number of sites in ΛN containing j particles in the configuration ω. We denote by N(ω) the sequence {Nj(ω),j∈Nb}. By definition, for every ω∈ΩN,b,m each site l∈ΛN is occupied by at least b particles, and Nω+≤m=m(N). It follows that AN,b,m is the range of N(ω) for ω∈ΩN,b,m; the two sums involving νj in (18) correspond to the two sums involving Nj(ω) in (1).Since the range ofN(ω) is AN,b,m, for ω∈ΩN,b,m the range of ΘN,b(ω) is the set of probability measures θN,b,ν whose components for j∈Nb have the form θN,b,ν;j=νj/N for ν∈AN,b,m. By (18) θN,b,ν takes values in PNb,c, the set of probability measures on Nb having mean c. It follows that the set(19)BN,b,m=θ∈PNb,c:θj=νjNforj∈Nbfor  someν∈AN,b,m is the range of ΘN,b(ω) for ω∈ΩN,b,m.In part (b) of the next theorem we state the local large deviation estimate for the event{ΘN,b=θN,b,ν}. In part (a) we introduce the Poisson distribution ρb,αb(c) that appears in the local estimate; ρb,αb(c) is defined in terms of a parameter αb(c) guaranteeing that it has mean c.In part (a) of TheoremC.2 in [7] we give the straightforward proof of the existence of αb(c) for b=1. The proof of the existence of αb(c) for general b∈N is much more subtle than the proof for b=1. The proof for general b∈N is given in Theorem A.2 in the present paper.Theorem 5.
(a) Fix a nonnegative integerb and a real number c∈(b,∞). For α∈(0,∞) let ρb,α be the measure on Nb having components ρb,α;j=[Zb(α)]-1·αj/j! for j∈Nb, where Z0,α=eα, and, for b∈N, Zb(α)=eα-∑j=0b-1αj/j!. Then there exists a unique value αb(c)∈(0,∞) such that ρb,αb(c) lies in the set PNb,c of probability measures on Nb having mean c. If b=0, then α0(c)=c. If b∈N, then αb(c) is the unique solution in (0,∞) of αZb-1(α)/Zb(α)=c.
(b) Fix a nonnegative integerb and a rational number c∈(b,∞). Let m be the function m(N) appearing in the definition of  ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. For any ν∈AN,b,m we define θN,b,ν∈PNb,c to have the components θN,b,ν;j=νj/N for j∈Nb. Then(20)1Nlog⁡PN,b,mΘN,b=θN,b,ν=-RθN,b,ν∣ρb,αbc+εNν.R(θN,b,ν∣ρb,αb(c)) is finite because it involves only finitely many components of θN,b,ν, and εN(ν)→0 uniformly for ν∈AN,b,m as N→∞.We now prove the local large deviation estimate in part (b) of Theorem5. This proof is based on a combinatorial argument that is reminiscent of and is as natural as the combinatorial argument used to prove Sanov’s theorem for empirical measures defined in terms of i.i.d. random variables having a finite state space [1, §3]. Part (b) of Theorem 5 is proved by analyzing the asymptotic behavior of the product of two multinomial coefficients that we now introduce.Givenν∈AN,b,m, our goal is to estimate the probability PN,b,m(ΘN,b=θN,b,ν), where θN,b,ν has the components θN,b,ν;j=νj/N for j∈Nb. A basic observation is that {ω∈ΩN,b,m:ΘN,b(ω)=θN,b,ν} coincides with (21)ΔN,b,m;ν=ω∈ΩN,b,m:Njω=νjforj∈Nb. It follows that (22)PN,b,mΘN,b=θN,b,ν=PN,b,mΔN,b,m;ν=1card⁡ΩN,b,m·card⁡ΔN,b,m;ν. Our first task is to determine the asymptotic behavior of card⁡(ΔN,b,m;ν). In determining the asymptotic behavior of card⁡(ΩN,b,m), we will use the fact that ΩN,b,m can be written as the disjoint union (23)ΩN,b,m=⋃ν∈AN,b,mΔN,b,m;ν.Letν∈AN,b,m be given. We start by expressing the cardinality of card⁡(ΔN,b,m;ν) as a product of two multinomial coefficients. For each configuration ω∈ΔN,b,m;ν, K particles are distributed onto the N sites of the lattice ΛN with j particles going onto νj sites for j∈Nb. We carry this out in two stages. In stage one K particles are placed into N bins, νj of which have j particles for j∈Nb. The number of ways of making this placement equals the multinomial coefficient K!/∏j∈Nb(j!)νj. This multinomial coefficient is well-defined since ∑j∈Nbjνj=K. Given this placement of K particles into N bins, the number of ways of moving the particles from the bins onto the sites 1,2,…,N of the lattice ΛN equals the multinomial coefficient N!/∏j∈Nbνj!. This second multinomial coefficient is well-defined since ∑j∈Nbνj=N. We conclude that the cardinality of ΔN,b,m;ν is given by the product of these two multinomial coefficients: (24)card⁡ΔN,b,m;ν=N!∏j∈Nbνj!·K!∏j∈Nbj!νj. Since ν+≤m, at most m of the components νj are positive. Such a product of multinomial coefficients is well known in combinatorial analysis [8, Thm. 2.10]. A related version of this formula is derived in Example III.23 of [34]. See also [35, p. 115] and formula (2) in [36, p. 36].The next two steps in the proof of the local estimate given in part (b) of Theorem5 are to prove the asymptotic formula for card⁡(ΔN,b,m;ν) in Lemma 6 and the asymptotic formula for card⁡(ΩN,b,m) in part (b) of Lemma 7. The proof of Lemma 6 is greatly simplified by a substitution in line 4 of (34). This substitution involves a parameter α∈(0,∞), which, we emphasize, is arbitrary in this lemma. The substitution in line 4 of (34) allows us to express the asymptotic behavior of both card(ΔN,b,m;ν) in Lemma 6 and card(ΩN,b,m) in Lemma 7 directly in terms of the relative entropy R(θN,b,ν∣ρb,α), where ρb,α is the probability measure on Nb having the components defined in part (a) of Theorem 5. One of the major issues in the proof of part (b) of Theorem 5 is to show that the arbitrary parameter α appearing in Lemmas 6 and 7 must take the value αb(c), which is the unique value of α guaranteeing that ρb,α∈PNb,c [Theorem 5(a)]. We show that α must equal αb(c) after the statement of Lemma 7.Lemma 6.
Fix a nonnegative integerb and a rational number c∈(b,∞). Let α be any real number in (0,∞), and let m be the function m(N) appearing in the definition of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. We define(25)fα,b,c,K=log⁡Zbα-clog⁡α+clog⁡K-c. For any ν∈AN,b,m, we define θN,b,ν∈PNb,c to have the components θN,b,ν;j=νj/N for j∈Nb. Then (26)1Nlog⁡card⁡ΔN,b,m;ν=-RθN,b,ν∣ρb,α+fα,b,c,K+ζNν.The quantity ζN(ν)→0 uniformly for ν∈AN,b,m as N→∞.Proof.
The proof is based on a weak form of Stirling’s approximation, which states that, for allN∈N satisfying N≥2 and for all n∈N satisfying 1≤n≤N, 1≤log⁡n!-(nlog⁡n-n)≤2log⁡N. We summarize the last formula by writing (27)log⁡n!=nlog⁡n-n+Olog⁡N,∀N∈N,N≥2,∀n∈1,2,…,N. The term denoted by O(log⁡N) satisfies 1≤O(log⁡N)≤2log⁡N.
To simplify the notation, we rewrite (24) in the form card⁡(ΔN,b,m;ν)=M1(N,ν)·M2(K,ν), where M1(N,ν) denotes the first multinomial coefficient on the right side of (24), and M2(K,ν) denotes the second multinomial coefficient on the right side of (24). We have (28)1Nlog⁡card⁡ΔN,b,m;ν=1Nlog⁡M1N,ν+1Nlog⁡M2K,ν.
The asymptotic behavior of the first term on the right side of the last display is easily calculated. Sinceν∈AN,b,m, there are ν+∈{1,2,…,m} positive components νj. Because of this restriction on the number ν+ of positive components of ν, we are able to control the error in line 3 of (29). We define ΨN(ν)={j∈Nb:νj≥1}. For each j∈ΨN(ν), since the components νj satisfy 1≤νj≤N, we have log⁡(νj!)=νjlog⁡νj-νj+O(log⁡N) for all N≥2. Using the fact that ∑j∈ΨN(ν)νj=N, we obtain(29)1Nlog⁡M1N,ν=1Nlog⁡N!-1N∑j∈ΨNνlog⁡νj!=1NNlog⁡N-N+Olog⁡N-1N∑j∈ΨNννjlog⁡νj-νj+Olog⁡N=-∑j∈NbθN,b,ν;jlog⁡θN,b,ν;j+ζN1-ζN2ν, where ζN(1)=[O(log⁡N)]/N→0 as N→∞ and ζN(2)(ν)=N-1∑j∈ΨN(ν)O(log⁡N). By the inequality noted after (27) and the fact that ν+≤m(30)0≤maxν∈AN,b,m⁡ζN2ν≤maxν∈AN,b,m⁡2N∑j∈ΨNνlog⁡N≤2mlog⁡NN. Since (mlog⁡N)/N→0 as N→∞, we conclude that ζN(2)(ν)→0 uniformly for ν∈AN,b,m as N→∞.
We now study the asymptotic behavior of the second term on the right side of (28). Since K=Nc, we obtain for all K≥2(31)1Nlog⁡M2K,ν=1Nlog⁡K!-1N∑j∈Nbνjlog⁡j!=clog⁡K-c-∑j∈NbθN,b,ν;jlog⁡j!+ζN3,where 0≤ζN3=O(log⁡K)/N=O(log⁡N)/N→0 as N→∞. The weak form of Stirling’s formula is used to rewrite the term log⁡(K!) in the last display, but not to rewrite the terms log⁡(j!), which we leave untouched.
Substituting (29) and (31) into (28), we obtain (32)1Nlog⁡card⁡ΔN,b,m;ν=1Nlog⁡M1N,ν+1Nlog⁡M2K,ν=-∑j∈NbθN,b,ν;jlog⁡θN,b,ν;jj!+clog⁡K-c+ζNν. In this formula ζN(ν)=ζN(1)-ζN(2)(ν)+ζN(3). As N→∞,(33)maxν∈AN,b,m⁡ζNν≤ζN1+maxν∈AN,b,m⁡ζN2ν+ζN3⟶0. We conclude that ζN(ν)→0 uniformly for ν∈AN,b,m as N→∞.
Now comes the key step, the purpose of which is to express the sum in the next-to-last line of (32) as the relative entropy R(θN,b,ν;j∣ρb,α), where α∈(0,∞) is arbitrary. To express the sum in the next-to-last line of (32) as R(θN,b,ν∣ρb,α), we rewrite the sum as shown in line 4 of the next display: (34)1Nlog⁡card⁡ΔN,b,m;ν=-∑j∈NbθN,b,ν;jlog⁡θN,b,ν;jj!+clog⁡K-c+ζNν=-∑j∈NbθN,b,ν;jlog⁡θN,b,ν;jαj/Zbα·j!·αjZbα+clog⁡K-c+ζNν=-RθN,b,ν∣ρb,α+log⁡Zbα-clog⁡α+clog⁡K-c+ζNν=-RθN,b,ν∣ρb,α+fα,b,c,K+ζNν. The facts that ∑j∈NbθN,b,ν;j=1 and ∑j∈NbjθN,b,ν;j=c are used to derive the next-to-last equality. The proof of Lemma 6 is complete.The next step in the proof of the local large deviation estimate in part (b) of Theorem5 is to prove the asymptotic formula for card⁡(ΩN,b,m) stated in part (b) of the next lemma. The proof of this lemma uses Lemma 6 in a fundamental way. After the statement of this lemma we show how to apply it and Lemma 6 to prove part (b) of Theorem 5.Lemma 7.
Fix a nonnegative integerb and a rational number c∈(b,∞). The following conclusions hold:(a)
limN→∞⁡N-1log⁡card⁡(AN,b,m)=0.(b)
Letα be the positive real number in Lemma 6, and let m be the function m(N) appearing in the definition of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. We define f(α,b,c,K)=log⁡Zb(α)-clog⁡α+clog⁡K-c. Then R(θ∣ρb,α) attains its infimum over θ∈PNb,c, and (35)1Nlog⁡card⁡ΩN,b,m=fα,b,c,K-minθ∈PNb,c⁡Rθ∣ρb,α+ηN. The quantity ηN→0 as N→∞.Before proving Lemma7, we derive the local large deviation estimate in part (b) of Theorem 5 by applying Lemmas 6 and 7. An integral part of the proof is to show how the arbitrary value of α∈(0,∞) appearing in these lemmas is replaced by the specific value αb(c) appearing in Theorem 5. As in the statement of part (b) of Theorem 5, let ν be any vector in AN,b,m and define θN,b,ν∈PNb,c to have the components θN,b,ν;j=νj/N for j∈Nb. By (22)(36)1Nlog⁡PN,b,mΘN,b=θN,b,ν=1Nlog⁡card⁡ΔN,b,m;ν-1Nlog⁡card⁡ΩN,b,m. Substituting the asymptotic formula for log⁡card(ΔN,b,m;ν) derived in Lemma 6 and the asymptotic formula for log⁡card(ΩN,b,m) given in part (b) of Lemma 7 yields (37)1Nlog⁡PN,b,mΘN,b=θN,b,ν=-RθN,b,ν∣ρb,α+minθ∈PNb,c⁡Rθ∣ρb,α+εNν. The error term εN(ν) equals ζN(ν)-ηN; ζN(ν) is the error term in Lemma 6, and ηN is the error term in Lemma 7. As N→∞, ζN(ν)→0 uniformly for ν∈AN,b,m, and ηN→0. It follows that εN(ν)→0 uniformly for ν∈AN,b,m as N→∞.We now consider the first two terms on the right side of (37). By part (b) of Theorem A.1 applied to θ=θN,b,ν∈PNb,c, for any α∈(0,∞)(38)RθN,b,ν∣ρb,α-minθ∈PNb,c⁡Rθ∣ρb,α=RθN,b,ν∣ρb,αbc.With this step we have succeeded in replacing the relative entropy R(θN,b,ν∣ρb,α) with respect to ρb,α, which appears in Lemma 6, by the relative entropy R(θN,b,ν∣ρb,αb(c)) with respect to ρb,αb(c), which appears in Theorem 5. Substituting the last equation into (37) gives(39)1Nlog⁡PN,b,mΘN,b=θN,b,ν=-RθN,b,ν∣ρb,αbc+εNν, where εN(ν)→0 uniformly for ν∈AN,b,m as N→∞. This is the conclusion of part (b) of Theorem 5.We now complete the proof of part (b) of Theorem5 by proving Lemma 7.Proof of Lemma7.
(a) We writeAN,b,m⊂⋃k=1m‍ν∈N0N:∑j∈Nb‍νj=N,ν+=k. By [8, Cor. 2.5] the number of elements in the set indexed by k equals the binomial coefficient C(N-1,k-1). Since by assumption m/N→0 as N→∞, for all sufficiently large N, the quantities C(N-1,k-1) are increasing and are maximal when k=m. Since C(N-1,k-1)≤C(N,k), it follows that(40)card⁡AN,b,m≤∑k=1mCN,k≤mCN,m=mN!m!N-m!.An application of the weak form of Stirling’s formula yields for all m≥2 and all N≥m+2(41)0≤1Nlog⁡card⁡AN,b,m≤log⁡mN-mNlog⁡mN-1-mNlog⁡1-mN+Olog⁡NN. Since m/N→0 as N→∞, we conclude that 0≤N-1log⁡card⁡(AN,b,m)→0 as N→∞. This completes the proof of part (a).
(b) The starting point is (23), which states that ΩN,b,m=⋃ν∈AN,b,m‍ΔN,b,m;ν. For distinct ν∈AN,b,m the sets ΔN,b,m;ν are disjoint. Hence (42)1Nlog⁡card⁡ΩN,b,m=1Nlog⁡∑ν∈AN,b,mcard⁡ΔN,b,m;ν=1Nlog⁡maxν∈AN,b,m⁡card⁡ΔN,b,m;ν+δN, where(43)0<δN=1Nlog⁡∑ν∈AN,b,mcard⁡ΔN,b,m;νmaxν∈AN,b,m⁡card⁡ΔN,b,m;ν≤1Nlog⁡card⁡AK,N,m. It follows from part (a) that δN→0 as N→∞.
We continue with the estimation ofcard⁡(ΩN,b,m). By Lemma 6(44)-minν∈AN,b,m⁡RθN,b,ν∣ρb,α+fα,b,c,K-maxν∈AN,b,m⁡ζNν≤1Nlog⁡maxν∈AN,b,m⁡card⁡ΔN,b,m;ν≤-minν∈AN,b,m⁡RθN,b,ν∣ρb,α+fα,b,c,K+maxν∈AN,b,m⁡ζNν.As proved in Lemma 6, maxν∈AN,b,m⁡ζNν→0 as N→∞. Hence by (42) (45)-minν∈AN,b,m⁡RθN,b,ν∣ρb,α+fα,b,c,K-maxν∈AN,b,m⁡ζNν+δN≤1Nlog⁡card⁡ΩN,b,m≤-minν∈AN,b,m⁡RθN,b,ν∣ρb,α+fα,b,c,K+maxν∈AN,b,m⁡ζNν+δN.Under the assumption that R(·∣ρb,α) attains its infimum over PNb,c, we define(46)ηN=1Nlog⁡card⁡ΩN,b,m-fα,b,c,K+minθ∈PNb,c⁡Rθ∣ρb,α. In the last two paragraphs of this proof, we show that ηN→0 as N→∞. Given this fact, the last equation yields the asymptotic formula (35) in part (b).
We now prove thatηN→0 as N→∞. To do this, we use (45) to write(47)ηN≤minν∈AN,b,m⁡RθN,b,ν∣ρb,α-minθ∈PNb,c⁡Rθ∣ρb,α+maxν∈AN,b,m⁡ζNν+δN. Like the second and third terms on the right side, the first term on the right side is nonnegative because AN,b,m is a subset of PNb,c. Since maxν∈AN,b,m⁡ζNν→0 and δN→0 as N→∞, it will follow that ηN→0 if we can show that R(·∣ρb,α) attains its infimum over PNb,c and that (48)limN→∞⁡minν∈AN,b,m⁡RθN,b,ν∣ρb,α=minθ∈PNb,c⁡Rθ∣ρb,α.
We now prove (48). R(·∣ρb,α) is lower semicontinuous on PNb [33, Lem. 1.4.3(b)] and thus on PNb,c. Since R(·∣ρb,α) has compact level sets in PNb,c [Theorem A.1(a)], it attains its infimum over PNb,c at some measure θ∗. We apply Theorem B.1 in [7] to θ=θ∗, obtaining a sequence θ(N) with the following properties: (1) for N∈N, θ(N)∈BN,b,m has components θj(N)=νj(N)/N for j∈Nb, where ν(N) is an appropriate sequence in AN,b,m; (2)θ(N)⇒θ∗ as N→∞; (3)R(θ(N)∣ρb,α)→R(θ∗∣ρb,α) as N→∞. The limit in (48) follows from the inequalities(49)minθ∈PNb,c⁡Rθ∣ρb,α≤minν∈AN,b,m⁡RθN,b,ν∣ρb,α≤RθN∣ρb,α and the limit R(θN∣ρb,α)→R(θ∗∣ρb,α)=minθ∈PNb,c⁡R(θ∣ρb,α) as N→∞. This completes the proof of Lemma 7 and thus the proof of the local estimate in part (b) of Theorem 5.In the next section we explain how the local large deviation estimate in part (b) of Theorem5 yields the LDP in Theorem 1.


## 4. Proof of Theorem1 from Part (b) of Theorem 5

In Theorem1 we state the LDP for the sequence ΘN,b of number-density measures. This sequence takes values in PNb,c, which is the set of probability measures on N having mean c∈(b,∞). The purpose of the present section is to explain how the local large deviation estimate in part (b) of Theorem 5 yields the LDP for ΘN,b. All details appear in Section 4 of [7]. The basic idea is first to prove the large deviation limit for ΘN,b lying in open balls in PNb,c and in other subsets defined in terms of open balls and then to use this large deviation limit to prove the LDP in Theorem 1.In Theorem8 we state the large deviation limit for open balls and other subsets defined in terms of open balls. Two types of open balls are considered. Let θ be a measure in PNb,c, and take r>0. Part (a) states the large deviation limit for open balls Bπ(θ,r)={μ∈PNb,c:π(θ,μ)<r}, where π denotes the Prohorov metric on PNb,c. This limit is used to prove the large deviation upper bound for compact subsets of PNb,c in part (b) of Theorem 1 and the large deviation lower bound for open subsets of PNb,c in part (d) of Theorem 1. Now let θ be a measure in PNb,[b,c]. Part (b) states the large deviation limit for sets of the form B^π(θ,r)∩PNb,c, where B^π(θ,r)={μ∈PNb,[b,c]:π(θ,μ)<r}. This limit is used to prove the large deviation upper bound for closed subsets in part (c) of Theorem 1. If θ∈PNb,c, then Bπ(θ,r)=B^π(θ,r)∩PNb,c, and the conclusions of parts (a) and (b) of the next theorem coincide.Theorem 8.
Fix a nonnegative integerb and a rational number c∈(b,∞). Let m be the function m(N) appearing in the definitions of ΩN,b,m in (3) and satisfying m(N)→∞ and m(N)2/N→0 as N→∞. The following conclusions hold:(a)
Letθ be a measure in PNb,c and take r>0. Then for any open ball Bπ(θ,r) in PNb,c, R(Bπ(θ,r)∣ρb,αb(c)) is finite, and one has the large deviation limit(50)limN→∞⁡1Nlog⁡PN,b,mΘN,b∈Bπθ,r=-RBπθ,r∣ρb,αbc.(b)
Letθ be a measure in PNb,[b,c] and take r>0. Then the set B^π(θ,r)∩PNb,c is nonempty, R(B^π(θ,r)∩PNb,c∣ρb,αb(c)) is finite, and one has the large deviation limit(51)limN→∞⁡1Nlog⁡PN,b,mΘN,b∈B^πθ,r∩PNb,c=-RB^πθ,r∩PNb,c∣ρb,αbc.We prove Theorem8 by applying the local large deviation estimate in part (b) of Theorem 5. A key step is to approximate probability measures in Bπ(θ,ε) and in B^π(θ,r)∩PNb,c by appropriate sequences of probability measures in the range of ΘN,b. This procedure allows one to show in part (a) that the infimum R(Bπ(θ,ε)∣ρb,αb(c)) can be approximated by the infimum of R(θ∣ρb,αb(c)) over θ lying in the intersection of Bπ(θ,ε) and the range of ΘN,b; a similar statement holds for the infimum in part (b). A set of hypotheses that allow one to carry out this approximation procedure is given in Theorem  4.2 in [7], a general formulation that yields Theorem 8 as a special case.Theorem1 states the LDP for the number-density measures ΘN,b. In order to complete the proof of Theorem 1, we must lift the large deviation limits in Theorem 8 to the large deviation upper bound for compact sets and for closed sets and the large deviation lower bound for open sets. The large deviation lower bound for open sets is immediate from the limit in part (a). To prove the large deviation upper bound for compact sets, we cover the compact set by open balls and use the limit in part (a); the large deviation upper bound for closed sets follows by a similar procedure involving part (b). The details of this procedure are carried out as an application of general formulation in Theorem  4.3 in [7].In the Appendix we prove two properties of the relative entropy and prove the existence of the quantityαb(c) appearing in part (a) of Theorem 5.



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