Large deviations for empirical measures of mean field Gibbs measures
aa r X i v : . [ m a t h . P R ] F e b LARGE DEVIATIONS FOR EMPIRICAL MEASURES OFMEAN-FIELD GIBBS MEASURES
WEI LIU AND LIMING WU
Abstract.
In this paper, we show that the empirical measure of mean-field modelsatisfies the large deviation principle with respect to the weak convergence topology orthe stronger Wasserstein metric, under the strong exponential integrability conditionon the negative part of the interaction potentials. In contrast to the known resultswe prove this without any continuity or boundedness condition on the interactionpotentials. The proof relies mainly on the law of large numbers and the exponentialdecoupling inequality of de la Pe˜na for U -statistics. AMS 2010 Subject classifications.
Primary: 60F10, 60K35; Secondary: 82C22.
Key words and Phrases.
Large deviations, empirical measure, U -statistics, interactingparticle systems, McKean-Vlasov equation, mean-field Gibbs measure.1. Introduction
In this paper, we consider a mean-field interacting system of n particles at positions x , · · · , x n in a separable and complete metric space ( S, ρ ) (say Polish space) confinedby a potential V : S → ( −∞ , + ∞ ]. The interaction between the particles is given bya family of Borel-measurable interaction potentials W ( k ) : S k → ( −∞ , + ∞ ] between k -particles, where 2 ≤ k ≤ N , and N ≥ n ≥ N ). The mean-fieldHamiltonian or energy functional H n : S n → ( −∞ , + ∞ ] is given by H n ( x , · · · , x n ) := n X i =1 V ( x i ) + n N X k =2 U n ( W ( k ) ) (1.1)where U n ( W ( k ) ) = 1 | I kn | X ( i , ··· ,i k ) ∈ I kn W ( k ) ( x i , · · · , x i k ) (1.2)is the U -statistic of order k , I kn := { ( i , · · · , i k ) ∈ N k | i , · · · , i k are different , ≤ i , · · · , i k ≤ n } and | I kn | denotes the number of elements in I kn (equal to n ! / ( n − k )!).The mean-field Gibbs probability measure P n on S n is defined by dP n ( x , · · · , x n ) := 1 Z n exp( − H n ( x , · · · , x n )) m ( dx ) · · · m ( dx n ) (1.3) The first author is supported by the CSC and NSFC(11731009, 11571262). where m is some nonnegative σ -finite measure on S equipped with the Borel σ -field B ( S ), and Z n := Z S · · · Z S exp( − H n ( x , · · · , x n )) m ( dx ) · · · m ( dx n ) (1.4)is the normalization constant (called partition function ).When N = 2, this model is called mean-field of pair interaction , and when N >
2, itis called mean-field of many-bodies interaction .The main objective of this paper is to study the large deviations of the empiricalmeasure L n ( x n ; · ) := 1 n n X i =1 δ x i ( · )of configuration x n = ( x , · · · , x n ), under the mean-field measure P n . We will simplydenote L n ( x n ; · ) by L n when there is no likelihood of confusion.In the case S = R d , P n is just the equilibrium state (or the invariant probabilitymeasure) of the system of n interacting particles described by: dX n ( t ) = √ d B t − ∇ H n ( X n ( t )) dt, (1.5)where X n ( t ) := ( X n ( t ) , · · · , X nn ( t )) T ( · T means the transposition ) takes values in ( R d ) n ,B t = ( B t , · · · , B nt ), B t , · · · , B nt are n independent Brownian motions taking values in R d . It is well-known that when n goes to infinity, L n ( X n ( t ); · ) converges to the solutionof the nonlinear McKean-Vlasov equation (the so-called propagation of chaos), underquite general condition [18].A classical problem is to establish conditions for the existence of a macroscopic limitof the empirical measures L n as the number of particles n → + ∞ . It is well-known thatthe large deviation principle (LDP in short) provides a strong exponential concentrationwith the speed n in terms of some explicit rate function, which is very useful for thestudy of the macroscopic limit and microscopic phenomena in statistical mechanics.In the case of pair interaction (i.e. N = 2), L´eonard established for the first time in[15, 1987] the LDP for the empirical measure L n under the Gibbs measure P n in theweighted weak convergence topology, when ν → RR W (2) ( x, y ) dν ( x ) dν ( y ) is continuousin some appropriate topology and it is bounded by some weighted function satisfying thestrong exponential integrability condition. By means of the weak convergence approachdeveloped in Dupuis and Ellis [9], Dupuis et al. established in [10] an LDP in the caseof pair interaction, by assuming that W (2) is lower bounded and lower semi-continuous(l.s.c. in short), which generalized the result obtained in [15]. For more results in thisfield the reader is referred to [1, 2, 5, 6, 12, 14, 17] and the references therein.The purpose of this paper is to establish the LDP for the empirical measures L n undera more general condition that W ( k ) , 2 ≤ k ≤ N are only measurable and their negativeparts W ( k ) , − satisfy the strong exponential integrability condition, which generalizesthe previous results in [10] and [15]. We first obtain the LDP with respect to the weakconvergence topology, then with respect to the Wasserstein metric by using Sanov’stheorem for the Wasserstein metric established by Wang et. al in [21]. Our main toolsare the law of large numbers (LLN in short) for the U -statistics and an exponentialinequality for U -statistics issued of de la Pe˜na decoupling inequality. ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 3
The paper is organized as follows. In the next section, we will first briefly introducesome notations and definitions concerning the LDP, and then present our main results.The proofs are presented in the third section.2.
Main result
Preliminaries.
We recall the definition of a rate function on a Polish space S andthe LDP for a sequence of probability measures on ( S, B ( S )). Definition 2.1 (Rate function) . I is said to be a rate function on S if it is a lowersemi-continuous function from S to [0 , ∞ ] (i.e., for all L ≥ , the level set [ I ≤ L ] isclosed). I is said to be a good rate function if it is inf-compact, i.e. [ I ≤ L ] is compactfor any L ∈ R . A consequence of a rate function being good is that its infimum is achieved over anynon-empty closed set.We denote by M ( S ) the space of probability measures on S . Definition 2.2 (LDP) . Let { ν n } n ∈ N be a sequence of probability measures in M ( S ) . ( a ) { ν n } n ∈ N is said to satisfy the large deviation lower bound with the speed n and arate function I if for any open subset G ∈ B ( S ) , l ( G ) := lim inf n → + ∞ n log ν n ( G ) ≥ − inf ν ∈ G I ( ν ); (2.1)( b ) { ν n } n ∈ N is said to satisfy the large deviation upper bound with the speed n and arate function I if for any closed subset F ∈ B ( S ) , u ( F ) := lim sup n → + ∞ n log ν n ( F ) ≤ − inf ν ∈ F I ( ν ); (2.2)( c ) { ν n } n ∈ N is said to satisfy the large deviation principle with the speed n and a ratefunction I if both (a) and (b) hold, and I is good. The LDP characterizes the exponential concentration behavior, as n → + ∞ , of asequence of probability measures { ν n } n ∈ N in terms of a rate function. This character-ization is via asymptotic upper and lower exponential bounds on the values that ν n assigns to measurable subsets of S .2.2. Main results.
Throughout this paper, we assume that C := Z S exp( − V ( x )) m ( dx ) < + ∞ . (2.3)Let α ( dx ) := 1 C exp( − V ( x )) m ( dx ) (2.4)be the probability measure on S , then the mean-field Gibbs probability measure P n canbe rewritten as dP n ( x , · · · , x n ) = 1 e Z n exp − n N X k =2 U n ( W ( k ) ) ! α ⊗ n ( dx , · · · , dx n ) , (2.5) WEI LIU AND LIMING WU where e Z n := Z n C n . Without interaction (i.e. W ( k ) = 0 for all k ), P n = α ⊗ n , i.e. the n particles are free and identically distributed with law α .Given a probability measure µ ∈ M ( S ), the relative entropy of ν with respect to µ is defined by H ( ν | µ ) = (cid:26) R S dνdµ ( x ) log dνdµ ( x ) µ ( dx ) , if ν ≪ µ ;+ ∞ , otherwise. (2.6)For any probability measure ν ∈ M ( S ) such that W ( k ) , − := ( − W ( k ) ) ∨ ∈ L ( ν ⊗ k )for all 2 ≤ k ≤ N , we define W ( k ) ( ν ) := Z S k W ( k ) ( x , · · · , x k ) dν ⊗ k ( x , · · · , x k ) ∈ ( −∞ , + ∞ ] . (2.7)The free energy of the state ν is given by H W ( ν ) := H ( ν | α ) + N X k =2 W ( k ) ( ν ) , if H ( ν | α ) < + ∞ and W ( k ) , − ∈ L ( ν ⊗ k ) , ≤ k ≤ N ;+ ∞ , otherwise. (2.8)Without loss of generality we may and will assume that W ( k ) is symmetric, i.e. W ( k ) ( x σ (1) , · · · , x σ ( k ) ) = W ( k ) ( x , · · · , x k )for any ( x , · · · , x k ) ∈ S k and any permutation σ on { , · · · , k } . We make the followingassumption on the interaction potentials ( W ( k ) ) ≤ k ≤ N : (A1) For each ≤ k ≤ N , the function W ( k ) : S k → ( −∞ , + ∞ ] is symmetric,measurable; its positive part W ( k ) , + satisfies H ( ν | α ) + Z S k W ( k ) , + ( x , · · · , x k ) dν ⊗ k ( x , · · · , x k ) < + ∞ for some ν ∈ M ( S ) (2.9) and its negative part W ( k ) , − satisfies the following strong exponential integrabilitycondition E [exp( λW ( k ) , − ( X , · · · , X k ))] < + ∞ , ∀ λ > where X , · · · , X k are i.i.d. random variables of the common law α defined in (2.4). Remark 2.3. (1) The simplest condition for (2.9) is: there is some measurable subset F of S with α ( F ) > F k W ( k ) , + is α ⊗ k -integrable. In fact one can take ν = hα , where the density h : S → R + is bounded, with support contained in F .(2) Under the exponential integrability condition (2.10), if H ( ν | α ) < + ∞ , then W ( k ) , − ∈ L ( ν ⊗ k ). In fact, for any λ > , by Donsker-Varadhan variational formula (see [9,Lemma 1.4.3.(a)] in the bounded case) and Fatou’s lemma (by approximating W ( k ) , − ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 5 with W ( k ) , − ∧ L , L ↑ + ∞ ), λ Z S k W ( k ) , − ( x , · · · , x k ) dν ⊗ k ( x , · · · , x k ) ≤ H ( ν ⊗ k | α ⊗ k ) + log Z S k e λW ( k ) , − ( x , ··· ,x k ) dα ⊗ k ( x , · · · , x k )= kH ( ν | α ) + log Z S k e λW ( k ) , − ( x , ··· ,x k ) dα ⊗ k ( x , · · · , x k ) < + ∞ . (3) When S = R d , N = 2 and R R d e − V dx < + ∞ , our assumption (A1) is satisfied inthe following two situations:(a) W (2) ( x, y ) = b | x − y | β with β < d , b > β = 1);(b) W (2) ( x, y ) = − b log | x − y | with b > R | x | p e − V dx < + ∞ for all p > W (2) , − satisfies the strong exponential integrability condition(2.10). Taking F = { x ∈ R d ; | x | ≤ R, V ( x ) ≥ − L } for R, L > F W (2) , + is dxdy -integrable, then α ⊗ -integrable. So the condition (2.9)is verified, by part (1) of this remark.Now we present our first main result, whose proof is given in the next section. Theorem 2.4.
Under assumption (A1) , H W is inf-compact on M ( S ) equipped withthe weak convergence topology, and − ∞ < inf µ ∈M ( S ) H W ( µ ) < + ∞ . (2.11) Moreover the sequence of probability measures { P n ( L n ∈ · ) } n ≥ N satisfies the LDP on M ( S ) equipped with the weak convergence topology, with speed n and the good ratefunction I W ( ν ) := H W ( ν ) − inf µ ∈M ( S ) H W ( µ ) , ν ∈ M ( S ) . (2.12) Furthermore, for any ν such that ν ≪ α and W ( k ) , − ∈ L ( ν ⊗ k ) , ∀ ≤ k ≤ N , the ratefunction I W ( ν ) can be identified as I W ( ν ) = lim n → + ∞ n H ( ν ⊗ n | P n ) . (2.13) Remark 2.5.
Notice that we have removed the lower semi-continuity and lower bound-ness conditions on W ( k ) in [10]. Our result generalizes the known results in L´eonard[15], Dupuis et al. [10]. Remark 2.6.
To see the main difficulty in this LDP, let us proceed naively in thecase of pair interaction: when W ( x, y ) := W (2) ( x, y ) is bounded and continuous, the U -statistic U n ( W ) = 1 n ( n − X ≤ i = j ≤ n W ( x i , x j ) WEI LIU AND LIMING WU is very close to RR W ( x, y ) L n ( dx ) L n ( dy ) which is continuous in L n in the weak conver-gence topology (see the proof of Lemma 3.7 below). So in that case the LDP followsfrom the LDP of L n under α ⊗ N (Sanov theorem) and Varadhan’s Laplace lemma, asshown by L´eonard [15]. When W is bounded and only measurable, we do not knowwhether the functional ν → RR W ( x, y ) ν ( dx ) ν ( dy ) is continuous in the (non-metrizable) τ -topology, whereas the Sanov theorem still holds in the τ -topology. The continuity ofthe last functional w.r.t. some appropriate topology is a basic assumption in [15]. Remark 2.7.
Since H W is inf-compact by Theorem 2.4, there is at least one minimizer.From the point of view of statistical physics, H W is an entropy or free energy associatedto the nonlinear McKean-Vlasov equation. The uniqueness of the minimizer meansthat there is no phase transition for the mean-field. Below we recall some works on theuniqueness in the case of pair interaction.For the uniqueness of the minimizer, it is sufficient to prove that H W is strictly convexalong some path ( ν t ) t ∈ [0 , connecting ν to ν , for any two probability measures ν , ν .Let ν ∗ be a minimizer of H W . Then the critical equation for the minimizer is ν ∗ ( dx ) = exp (cid:0) − V ( x ) − π ν ∗ W (2) ( x ) (cid:1) m ( dx ) /C, (2.14)where π ν ∗ W (2) ( x ) := Z S W (2) ( x, y ) ν ∗ ( dy ) , and C := Z S exp (cid:0) − V ( x ) − π ν ∗ W (2) ( x ) (cid:1) m ( dx )is the normalization constant.If S = R d , the critical equation above is equivalent to the following stationary equa-tion of the nonlinear McKean-Vlasov equation: △ ν ∗ + ∇ · ( ν ∗ ∇ V ) + ∇ · [(2 ∇ W (2) ∗ ν ∗ ) · ν ∗ ] = 0 , (2.15)where the symbols ∇ and ∇· denote the gradient operator and divergence operatorrespectively. For the uniqueness of the solution of (2.15), the reader is referred to Mc-Cann [16] and Carrilo et al. [4]. These authors showed that H W is strictly displacementconvex (i.e. along the W -geodesic) under various sufficient conditions on the convexityof the confinement potential V and the pair interaction potential W (2) .We also consider M ( S ) equipped with the Wasserstein topology, which is muchstronger than the weak convergence topology. The L p -Wasserstein distance ( p ≥ ρ on S , between any two probability measures µ and ν on S , is defined by W p ( µ, ν ) = inf ξ ∈ Π( µ,ν ) (cid:18)Z Z S × S ρ p ( x, y ) ξ ( dx, dy ) (cid:19) /p , (2.16)where Π( µ, ν ) is the set of all probability measures on S × S with marginal distribution µ and ν respectively (say couplings of µ and ν ). ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 7
The
Wasserstein space of order p is defined as M p ( S ) = (cid:26) µ ∈ M ( S ); Z S ρ p ( x, x ) µ ( dx ) < + ∞ (cid:27) , where x is some fixed point of S . It is known that W p is a finite distance on M p ( S )and ( M p ( S ) , W p ) is a Polish space (see Villani [19, 20]). Theorem 2.8.
Assume Z S exp { λρ p ( x, x ) } α ( dx ) < + ∞ , ∀ λ > , (2.17) for some (hence for any) x ∈ S . Under the assumption (A1) , the sequence of prob-ability measures { P n ( L n ∈ · ) } n ≥ N satisfies the LDP on ( M p ( S ) , W p ) with speed n andthe good rate function I W defined in (2.12). Remark 2.9.
In [10], Dupuis et al. imposed the following non-explicit condition forthe LDP result above when S = R d and N = 2: there exists a lower-semicontinuousfunction φ : R + → R with lim s → + ∞ φ ( s ) s = + ∞ , such that for every µ ∈ M ( R d ), Z R d φ ( | x | p ) µ ( dx ) ≤ inf ξ ∈ Π( µ,µ ) (cid:26) H ( ξ | α ⊗ ) + Z R d × R d W ( x, y ) ξ ( dxdy ) (cid:27) . Proof of the main results
Let P ∗ n be the measure by removing the normalization constant e Z n from P n presentedin (2 . dP ∗ n ( x , · · · , x n ) := exp − n N X k =2 U n ( W ( k ) ) ! α ⊗ n ( dx , · · · , dx n ) . (3.1)3.1. Large deviation (LD in short) lower bound for P ∗ n . First we present thelaw of large numbers of the U -statistic (see [13, Corollary 3.1.1]). Let X , X , · · · be asequence of i.i.d. random variables in a measurable space ( S, B ( S )) . Let Φ : S k → R bea symmetric and measurable function of k ( k ≥
2) variables.
Lemma 3.1. [13, Korolyuk and Borovskich]
Assume that E | Φ( X , · · · , X k ) | < + ∞ , (3.2) then U n (Φ) → E Φ( X , · · · , X k ) (3.3) as n → + ∞ with probability . WEI LIU AND LIMING WU
Proof.
For the sake of completeness, we re-present the simple proof in [13].Let Π n be the set of all permutations of { , · · · , n } and B n the σ -algebra defined by B n := σ (cid:8) B n × C n | C n ∈ B ( S [ n +1 , + ∞ ) ) , B n ∈ B ( S n ) , π B n = 1 B n , ∀ π ∈ Π n (cid:9) . The σ -algebra B n remains unchanged under any permutation in Π n , and B n ⊇ B n +1 for every n ≥ . For any ( i , · · · , i k ) ∈ I kn , by (3.2) we have E [Φ( X i , · · · , X i k ) | B n ] = E [Φ( X , · · · , X k ) | B n ] , which yields U n (Φ) = E [Φ( X , · · · , X k ) | B n ] . According to the limit theorem for reversed martingales and the 0-1 law for B ∞ = T n ≥ B n , U n (Φ) a.s. −−→ E [Φ( X , · · · , X k ) | B ∞ ] = E Φ( X , · · · , X k ) . (cid:3) We have the following LD lower bound for the empirical measure L n under P ∗ n . Proposition 3.2.
Without any integrability condition on ( W ( k ) ) ≤ k ≤ N , the followinglarge deviation lower bound holds for { P ∗ n { L n ∈ ·}} n ≥ N : l ∗ ( G ) := lim inf n → + ∞ n log P ∗ n { L n ∈ G } ≥ − inf { H W ( ν ) | ν ∈ G, W ( k ) ∈ L ( ν ⊗ k ) , ≤ k ≤ N } , (3.4) for any open subset G of M ( S ) . In particular, we have lim inf n → + ∞ n log e Z n ≥ − inf { H W ( ν ) | ν ∈ M ( S ) , W ( k ) ∈ L ( ν ⊗ k ) , ≤ k ≤ N } . (3.5) Proof.
Since (3.5) is obtained just by taking G as M ( S ) in (3.4), we only need to prove(3.4). For (3.4), it is enough to show that for any ν ∈ G such that H ( ν | α ) < + ∞ and W ( k ) ∈ L ( ν ⊗ k ) , ≤ k ≤ N , l ∗ ( G ) ≥ − H W ( ν ) . Let N ( ν, δ ) be the open ball centered at ν in M ( S ) with radius δ in the L´evy-Prokhorov metric d w such that N ( ν, δ ) ⊂ G . Introduce the events A n := { x n = ( x , · · · , x n ) ∈ S n | L n = L n ( x n , · ) ∈ N ( ν, δ ) } ,B n := { ( x , · · · , x n ) | n n X i =1 log dνdα ( x i ) ≤ H ( ν | α ) + ε } ,C n := { ( x , · · · , x n ) | N X k =2 U n ( W ( k ) ) ≤ N X k =2 W ( k ) ( ν ) + ε } . ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 9
Then we have for any ε > ,P ∗ n { L n ∈ N ( ν, δ ) } ≥ Z A n (cid:18) dν ⊗ n dP ∗ n ( x , · · · , x n ) (cid:19) − dν ⊗ n ( x , · · · , x n )= Z A n exp − n X i =1 log dνdα ( x i ) ! exp − n N X k =2 U n ( W ( k ) ) ! ν ⊗ n ( dx , · · · , dx n ) ≥ ν ⊗ n ( A n ∩ B n ∩ C n ) exp − n [ H ( ν | α ) + ε ] − n [ N X k =2 W ( k ) ( ν ) + ε ] ! = ν ⊗ n ( A n ∩ B n ∩ C n ) exp ( − nH W ( ν ) − nε ) , (3.6)We claim that lim n → + ∞ ν ⊗ n ( A n ∩ B n ∩ C n ) = 1 . Indeed, by the LLN, it is obviousthat ν ⊗ n ( A n ) → ν ⊗ n ( B n ) → n → + ∞ . By the LLN of U -statistics in Lemma 3.1, we also havelim n → + ∞ ν ⊗ n ( C n ) = 1 . With this claim in hand, we immediately get from (3.6) that l ∗ ( G ) ≥ lim inf n → + ∞ n log P ∗ n { L n ∈ N ( ν, δ ) } ≥ − H W ( ν ) − ε, (3.7)which completes the proof since ε > (cid:3) Decoupling inequality of de la Pe˜na and the key lemma.
We first recallthe decoupling inequality of de la Pe˜na [7, 1992].
Proposition 3.3. [7, de la Pe˜na]
Let { X i } i ≥ be a family of i.i.d. random variables ina measurable space ( S, B ( S )) and suppose that ( X j , · · · , X jn ) kj =1 are independent copiesof ( X , · · · , X n ) . Let Ψ be any convex increasing function on [0 , + ∞ ) . Let Φ : S k → R be a symmetric and measurable function of k variables such that E | Φ( X , · · · , X k ) | < + ∞ , (3.8) then E Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( i ,i , ··· ,i k ) ∈ I kn Φ( X i , · · · , X i k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E Ψ C k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( i ,i , ··· ,i k ) ∈ I kn Φ( X i , · · · , X ki k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.9) where C = 8 and C k = 2 k k Y j =2 ( j j − for k > . Besides the decoupling inequality of de la Pe˜na above, we also require the followinginequality.
Lemma 3.4.
Let ≤ k ≤ n , and { X ji ; 1 ≤ i ≤ n, ≤ j ≤ k } be independent randomvariables. For any ( i , · · · , i k ) ∈ I kn , let Φ i , ··· ,i k : S k → R be a measurable function of k variables, then log E exp ( n − k )! n ! X ( i , ··· ,i k ) ∈ I kn Φ i , ··· ,i k ( X i , · · · , X ki k ) ≤ ( n − k + 1)! n ! X ( i , ··· ,i k ) ∈ I kn log E exp (cid:18) n − k + 1 Φ i , ··· ,i k ( X i , · · · , X ki k ) (cid:19) . (3.10) Proof.
For k = 1, (3.10) is obviously an equality. Next we prove this lemma by induc-tion. Assume that (3.10) is valid for k −
1. Denote the left-hand side of (3.10) by B k and write P ( i , ··· ,i k − ) ∈ I k − n as P I k − n for simplicity. We have B k = log E X k { E [exp( ( n − k + 1)! n ! X I k − n X i k / ∈{ i , ··· ,i k − } n − k + 1 Φ i , ··· ,i k ( X i , · · · , X ki k )) | X k ] } . (3.11)Given X k = ( X k , · · · , X kn ), let˜Φ i , ··· ,i k − := 1 n − k + 1 X i k : i k / ∈{ i , ··· ,i k − } Φ i , ··· ,i k ( X i , · · · , X k − i k − , X ki k ) . (3.12)By the assumption of ( k − th step, we get B k ≤ log E X k exp ( n − k + 2)! n ! X I k − n log E [exp( 1 n − k + 2 ˜Φ i , ··· ,i k − ) | X k ] = log E X k exp ( n − k + 1)! n ! X I k − n log { E [exp( 1 n − k + 2 ˜Φ i , ··· ,i k − ) | X k ] } n − k +2 ≤ ( n − k + 1)! n ! X I k − n log E X k ((cid:20) E exp (cid:18) n − k + 2 ˜Φ i , ··· ,i k − (cid:19) | X k (cid:21) n − k +2 ) (3.13)where the last inequality follows by the convexity of X → log E e X (a consequence ofH¨older’s inequality). ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 11
Next we deal with the logarithmic term in the last inequality above. Given ( i , · · · , i k − ), E X k { [ E exp( 1 n − k + 2 ˜Φ i , ··· ,i k − ) | X k ] n − k +2 } = E X k { [ E exp( 1( n − k + 2)( n − k + 1) X i k / ∈{ i , ··· ,i k − } Φ i , ··· ,i k ( X i , · · · , X k − i k − , X ki k )) | X k ] n − k +2 }≤ E X k ((cid:20) Π i k : i k / ∈{ i , ··· ,i k − } E [exp( 1 n − k + 2 Φ i , ··· ,i k ( X i , · · · , X k − i k − , X ki k )) | X k ] (cid:21) n − k +2 n − k +1 ) (by H¨older’s inequality) ≤ E X k (cid:26) Π i k : i k / ∈{ i , ··· ,i k − } E [exp( 1 n − k + 1 Φ i , ··· ,i k ( X i , · · · , X k − i k − , X ki k )) | X k ] (cid:27) (by Jensen’s inequality)= Π i k : i k / ∈{ i , ··· ,i k − } E exp( 1 n − k + 1 Φ i , ··· ,i k ( X i , · · · , X k − i k − , X ki k ))(by the independence of X k , · · · , X kn )(3.14)Plugging (3.14) into (3.13), we get the desired inequality (3.10). (cid:3) Let { X i } i ≥ be a family of i.i.d. random variables of law α . Denote by Λ n ( · ; W ( k ) )the logarithmic moment generating function associated with the U -statistic of order k ,i.e., for any n ≥ k ≥ λ > , Λ n ( λ ; W ( k ) ) := 1 n log E [exp( λnU n ( W ( k ) ))] . (3.15)Now we present our key lemma, which is a corollary of the decoupling inequality of dela Pe˜na and Lemma 3.4. Lemma 3.5. If E | W ( k ) ( X , · · · , X k ) | < + ∞ , then for any ≤ k ≤ n and λ > , Λ n ( λ ; W ( k ) ) ≤ k log E [exp( kC k λ | W ( k ) ( X , · · · , X k ) | )] , (3.16) where C k is defined as in Proposition 3.3.Proof. Let ( X j , · · · , X jn ) kj =1 be independent copies of ( X , · · · , X n ). By Lemma 3.3 (thedecoupling inequality of de la P˜ena) and Lemma 3.4, taking Φ i , ··· ,i k ≡ W ( k ) for any ( i , · · · , i k ) ∈ I kn , we get for any λ > n ( λ ; W ( k ) ) = 1 n log E [exp( λn ( n − k )! n ! X ( i , ··· ,i k ) ∈ I kn W ( k ) ( X i , · · · , X i k ))] ≤ n log E [exp( ( n − k )! n ! X ( i , ··· ,i k ) ∈ I kn λnC k | W ( k ) | ( X i , · · · , X ki k )] (by (3.9)) ≤ n ( n − k + 1)! n ! X ( i , ··· ,i k ) ∈ I kn log E exp( λnC k n − k + 1 | W ( k ) | ( X i , · · · , X ki k )) (by (3.10))= n − k + 1 n log E exp( λnC k n − k + 1 | W ( k ) | ( X , · · · , X k )) ≤ k log E exp( kC k λ | W ( k ) | ( X , · · · , X k )) , (3.17)where the last inequality follows by the non-decreasingness of a → a log E e aX on(0 , + ∞ ) and the fact that nn − k +1 ≤ k for all n ≥ k . (cid:3) We have the following exponential approximation of the U -statistics. Lemma 3.6.
Assume that for any λ > , E [exp( λ | W ( k ) | ( X , · · · , X k ))] < + ∞ . (3.18) Then there exists a sequence of bounded continuous functions { W ( k ) m } m ≥ such that forany δ > , lim m → + ∞ lim sup n → + ∞ n log P {| U n ( W ( k ) ) − U n ( W ( k ) m ) | > δ } = −∞ . (3.19) Proof.
For any function W ( k ) satisfying (3.18) and integer m ∈ N ∗ , considering thetruncation W ( k ) ,L := ( − L ) ∨ ( W ( k ) ∧ L ), we have by dominated convergence thatlog E [exp( m | W ( k ) − W ( k ) ,L | ( X , · · · , X k ))] → L → + ∞ . Then we can choose L = L ( m ) so thatlog E [exp( m | W ( k ) − W ( k ) ,L ( m ) | ( X , · · · , X k ))] ≤ m . For m ∈ N ∗ and L = L ( m ) > { W ( k ) ,Ll } l ≥ on S k such that W ( k ) ,Ll ( X , · · · , X k ) → W ( k ) ,L ( X , · · · , X k ) in L as l goes to infinity, and | W ( k ) ,Ll ( X , · · · , X k ) | ≤ L (otherwise considering the truncation( − L ) ∨ ( W ( k ) ,Ll ∧ L )) for all l ≥
1. Sinceexp( m [ | W ( k ) − W ( k ) ,L | ( X , · · · , X k ) + | W ( k ) ,L − W ( k ) ,Ll | ( X , · · · , X k )]) ≤ exp( m | W ( k ) − W ( k ) ,L | ( X , · · · , X k ) + 2 mL ) ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 13 for any l ≥
1, we have by the dominated convergence E [exp( m [ | W ( k ) − W ( k ) ,L | ( X , · · · , X k ) + | W ( k ) ,L − W ( k ) ,Ll | ( X , · · · , X k )])] → E [exp( m | W ( k ) − W ( k ) ,L | ( X , · · · , X k ))]as l → ∞ . Thus for L = L ( m ), we can find l = l ( m ) such thatlog E [exp( m [ | W ( k ) − W ( k ) ,L | ( X , · · · , X k ) + | W ( k ) ,L − W ( k ) ,Ll | ( X , · · · , X k )])] ≤ m . Setting W ( k ) m = W ( k ) ,L ( m ) l ( m ) which is bounded and continuous, we have by the triangularinequality log E [exp( m | W ( k ) − W ( k ) m | ( X , · · · , X k ))] ≤ m . For any λ >
0, sincelog E [exp( λ | W ( k ) − W ( k ) m | ( X , · · · , X k ))] ≤ λm log E [exp( m | W ( k ) − W ( k ) m | ( X , · · · , X k ))]for m ≥ λ by Jensen’s inequality, we see that for the sequence of bounded and continuousfunctions { W ( k ) m } m ≥ ,log E [exp( λ | W ( k ) − W ( k ) m | ( X , · · · , X k ))] → , (3.20)as m → + ∞ .For any δ, λ > , by Chebyshev’s inequality we have P {| U n ( W ( k ) ) − U n ( W ( k ) m ) | > δ } ≤ e − nλδ E [exp( λnU n ( | W ( k ) − W ( k ) m | ))] . (3.21)Applying Lemma 3.5, we get1 n log P {| U n ( W ( k ) ) − U n ( W ( k ) m ) | > δ } ≤ − λδ + 1 k log E [exp( kC k λ | W ( k ) − W ( k ) m | ( X , · · · , X k ))] . (3.22)Letting m → + ∞ and using (3.20), we get the desired result (3.19), since λ > (cid:3) LDP of U -statistics. We begin with
Lemma 3.7.
Let ( X n ) n ≥ be a sequence of i.i.d. random variables valued in S , of com-mon law α . Assume that ( W ( k ) ) ≤ k ≤ N are measurable and satisfy the strong exponentialintegrability condition E (cid:2) exp (cid:0) λ | W ( k ) ( X , · · · , X k ) | (cid:1)(cid:3) < + ∞ , ∀ λ > , (3.23) then { P (( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )) ∈ · ) } n ≥ N satisfies the LDP on the product space M ( S ) × R N − , with good rate function I defined by I ( ν, z , · · · , z N ) := (cid:26) H ( ν | α ) , if z k = W ( k ) ( ν ) for all ≤ k ≤ N + ∞ , otherwise (3.24) for any ( ν, z , · · · , z N ) ∈ M ( S ) × R N − . Proof.
Let { W ( k ) m } m ≥ , ≤ k ≤ N be the sequences of bounded continuous functions asin Lemma 3.6 such that for any λ > ε ( λ, m, k ) := log Z S k exp( λ | W ( k ) m − W ( k ) | ) dα ⊗ k → , as m → + ∞ . (3.25)For any m ≥
1, let f m ( ν ) := (cid:18) ν, Z S W (2) m dν ⊗ , · · · , Z S N W ( N ) m dν ⊗ N (cid:19) ,f ( ν ) := (cid:18) ν, Z S W (2) dν ⊗ , · · · , Z S N W ( N ) dν ⊗ N (cid:19) and consider the following metric on the product space M ( S ) × R N − : d (( ν , z , · · · , z N ) , (˜ ν , ˜ z , · · · , ˜ z N )) := d w ( ν , ˜ ν ) + N X k =2 | z k − ˜ z k | . Note that d ( f m ( ν ) , f ( ν )) = P Nk =2 (cid:12)(cid:12)(cid:12)R S k ( W ( k ) m − W ( k ) ) dν ⊗ k (cid:12)(cid:12)(cid:12) . Below we separate the proofof the LDP into three points.
1) The continuity of f m ( ν ) . We first prove the continuity of f m ( ν ) or equiva-lently that of ν → R W ( k ) m dν ⊗ k on M ( S ) in the weak convergence topology for each k = 2 , · · · , N . In fact let ν n → ν in ( M ( S ) , d w ). By Skorokhod’s representation theo-rem (see [3, Theorem 6.7]), one can construct a sequence of S -valued random variables Y n of law ν n , converging a.s. to Y of law ν . Let ( Y ( i ) n , n ≥ Y ( i ) ) ≤ i ≤ k be indepen-dent copies of ( Y n , n ≥ Y ). Then Y ( i ) n → Y ( i ) , a.s. too for each i , in other words,( Y (1) n , · · · , Y ( k ) n ) → ( Y (1) , · · · , Y ( k ) ) , a.s. . Thus ν ⊗ kn → ν ⊗ k weakly on S k , which impliesthe continuity of the functional above.
2) Exponentially good approximation of ( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )) by f m ( L n ) . By (3.19) in Lemma 3.6, for any δ > n →∞ n log P (cid:0) d (( L n , U n ( W (2) m ) , · · · , U n ( W ( N ) m )) , ( L n , U n ( W (2) ) , · · · , U n ( W ( N ) ))) > δ (cid:1) → −∞ as m → + ∞ , i.e. ( L n , U n ( W (2) m ) , · · · , U n ( W ( N ) m )) is an exponentially good approxima-tion of ( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )).Moreover ( L n , U n ( W (2) m ) , · · · , U n ( W ( N ) m )) and f m ( L n ) are exponentially equivalent, be-cause the following uniform estimate holds: | U n ( W ( k ) m ) − Z W ( k ) m dL ⊗ kn | ≤ | U n ( W ( k ) m ) − | I kn | n k U n ( W ( k ) m ) | + (1 − | I kn | n k ) k W ( k ) m k ∞ ≤ (cid:18) − | I kn | n k (cid:19) k W ( k ) m k ∞ → n → ∞ . Then as m → ∞ , f m ( L n ) is an exponentially good approximation of( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )). ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 15
Therefore by Sanov’s theorem (the LDP of L n ) and the theorem of approximationof LDP ([8, Theorem 4.2.23]), for the desired LDP it remains to show that for any L > ν : H ( ν | α ) ≤ L d ( f m ( ν ) , f ( ν )) → , as m → + ∞ . (3.26)In fact for any λ > , L > ν with H ( ν | α ) ≤ L , we have by Donsker-Varadhanvariational formula (see [9, Lemma 1.4.3.(a)]) and Fatou’s lemma, Z S k | W ( k ) m − W ( k ) | dν ⊗ k ≤ λ (cid:18) H ( ν ⊗ k | α ⊗ k ) + log Z S k exp( λ | W ( k ) m − W ( k ) | ) dα ⊗ k (cid:19) ≤ λ [ kL + ε ( λ, m, k )] , ≤ k ≤ N. (3.27)where (3.26) follows for λ > m →∞ ε ( λ, m, k ) = 0 for any λ > (cid:3) Proof of Theorem 2.4.
Proof of Theorem 2.4.
We divide its proof into three steps.
Step 1. W ( k ) is upper bounded. In this upper bounded case, E e λ | W ( k ) | ( X , ··· ,X k ) < + ∞ for all λ > (A1) . By Lemma 3.7, under P = α ⊗ N ∗ , ( L n , U n ( W (2) ) , · · · , U n ( W ( N ) ))satisfies the LDP on M ( S ) × R N − with the rate function I ( ν, z , · · · , z N ) given by(3.24). Since P Nk =2 U n ( W ( k ) ) is continuous in ( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )), andlim sup n →∞ n log E exp − np N X k =2 U n ( W ( k ) ) ! < + ∞ for some p > p >
1) by Lemma 3.5 and (A1) , we can apply the tiltedLDP (Ellis [11, Theorem II.7.2.]) to conclude that P n (( L n , U n ( W (2) ) , · · · , U n ( W ( N ) )) ∈· ) satisfies the LDP, with the rate function ˜ I given by˜ I ( ν, z , · · · , z N ) = I ( ν, z , · · · , z N ) + N X k =2 z k − inf ( ν,z , ··· ,z N ) [ I ( ν, z , · · · , z N ) + N X k =2 z k ]= H ( ν | α ) + P Nk =2 W ( k ) ( ν ) − inf ν ′ ∈M ( S ) ,H ( ν ′ | α ) < + ∞ { H ( ν ′ | α ) + P Nk =2 W ( k ) ( ν ′ ) } if H ( ν | α ) < + ∞ , z k = W ( k ) ( ν ) , ≤ k ≤ N + ∞ , otherwise.= ( H W ( ν ) − inf M ( S ) H W if H ( ν | α ) < + ∞ , z k = W ( k ) ( ν ) , ≤ k ≤ N + ∞ otherwise.Hence P n ( L n ∈ · ) satisfies the LDP with the rate function I W , by the contractionprinciple. Notice that H W = I W + inf M ( S ) H W is inf-compact, as I W . Step 2. General unbounded case.
For any
L >
0, let W ( k ) L := W ( k ) ∧ L . Then H W L ( ν ) = H ( ν | α ) + N X k =2 Z S k W ( k ) L dν ⊗ k , if H ( ν | α ) < + ∞ and + ∞ otherwiseis inf-compact on M ( S ), by Step 1. Therefore H W is also inf-compact since H W L ( ν ) ↑ H W ( ν ) as L ↑ + ∞ .For any closed subset C of M ( S ) , we have for any L > P ∗ n { L n ∈ C } = Z S n { L n ∈ C } exp − n N X k =2 U n ( W ( k ) ) ! dα ⊗ n ≤ Z S n { L n ∈ C } exp − n N X k =2 U n ( W ( k ) L ) ! dα ⊗ n ≤ exp (cid:18) − n inf ν ∈ C H W L ( ν ) + o ( n ) (cid:19) , (3.28)where the last inequality follows from Lemma 3.7 and Varadhan’s Laplace lemma.Hence we get lim sup n → + ∞ n log P ∗ n { L n ∈ C } ≤ − inf ν ∈ C H W L ( ν ) , which leads to lim sup n → + ∞ n log P ∗ n { L n ∈ C } ≤ − inf ν ∈ C H W ( ν ) (3.29)because inf ν ∈ C H W L ( ν ) ↑ inf ν ∈ C H W ( ν ) as L ↑ + ∞ by the inf-compactness of H W L , H W .Taking C = M ( S ) in (3.29) we obtainlim sup n → + ∞ n log e Z n ≤ − inf ν ∈M ( S ) H W ( ν ) . By the lower bound (3.5) in Proposition 3.2 and the upper bound above, and the factthat H W ( ν ) = + ∞ once if W ( k ) / ∈ L ( ν ⊗ k ) for some 2 ≤ k ≤ N , we obtainlim n →∞ n log ˜ Z n = − inf ν ∈M ( S ) H W ( ν ) (3.30)which is a finite quantity (i.e. in R ) by the inf-compactness of H W and (2.9). With thiskey equality (3.30) in hand, the LDP of { P n ( L n ∈ · ) } n ≥ N follows from the lower boundin Proposition 3.2 and the upper bound (3.29). Step 3.
Finally it remains to show the identification (2.13) of the rate function I W .For any ν such that ν ≪ α and W ( k ) , − ∈ L ( ν ⊗ k ) , ≤ k ≤ N , we have1 n H ( ν ⊗ n | P n ) = 1 n E ν ⊗ n log dν ⊗ n dα ⊗ n + n N X k =2 U n ( W ( k ) ) + log ˜ Z n ! = H ( ν | α ) + N X k =2 Z W ( k ) dν ⊗ n + 1 n log ˜ Z n ARGE DEVIATIONS FOR MEAN-FIELD GIBBS MEASURES 17 which yields (2.13) by (3.30), as n → + ∞ . (cid:3) Proof of Theorem 2.8.
We first present the result of Sanov’s theorem in theWasserstein distance by Wang et. al. [21].
Proposition 3.8.
Let { X n } n ≥ be a sequence of i.i.d. random variables defined ona probability space (Ω , F , P ) with values in a Polish space ( S, ρ ) , of common law α ,then { P ( L n ∈ · ) } n ≥ satisfies the LDP on ( M p ( S ) , W p ) with speed n and the good ratefunction H ( ·| α ) , if and only if Z S exp { λρ p ( x, x ) } α ( dx ) < + ∞ , ∀ λ > for some (hence for any) x ∈ S .Proof of Theorem 2.8. Since we have established the LDP of the empirical measure L n under P n on M ( S ) equipped with the weak convergence topology, it is sufficient toprove the exponential tightness of { P n ( L n ∈ · ) } n ≥ N in ( M p ( S ) , W p ) (see [8, Corollary4.2.6]).For any compact subset K ⊂ M p ( S ), and any fixed a, b ∈ (1 , + ∞ ) with a + b = 1,we have by H¨older’s inequality P n { L n / ∈ K } = 1 e Z n Z { L n / ∈ K } exp − n N X k =2 U n ( W ( k ) ) ! dα ⊗ n ≤ e Z n (cid:2) α ⊗ n { L n / ∈ K } (cid:3) /a × Z exp − bn N X k =2 U n ( W ( k ) ) ! dα ⊗ n ! /b . (3.31)Hence we havelim sup n → + ∞ n log P n { L n / ∈ K }≤ a lim sup n → + ∞ n log α ⊗ n { L n / ∈ K } − lim sup n → + ∞ n log e Z n + 1 b lim sup n → + ∞ n log Z exp − n N X k =2 U n ( bW ( k ) ) ! dα ⊗ n = 1 a lim sup n → + ∞ n log α ⊗ n { L n / ∈ K } + inf ν ∈M ( S ) H W ( ν ) − b inf ν ∈M ( S ) H bW ( ν ) , (3.32)where inf ν ∈M ( S ) H W ( ν ) andinf ν ∈M ( S ) H bW ( ν ) := inf ν ∈M ( S ) { H ( ν | α ) + N X k =2 Z S k bW ( k ) dν ⊗ k } are finite by (2.11) in Theorem 2.4. Note that under the exponential integrability condition (2.17), the LDP holds for L n under α ⊗ n with respect to the Wasserstein topology by Proposition 3.8. Thus for any L >
0, there exists a compact subset K L ⊂ M p ( S ) such thatlim sup n → + ∞ n log α ⊗ n { L n / ∈ K L } ≤ − aL − a inf ν ∈M ( S ) H W ( ν ) + ab inf ν ∈M ( S ) H bW ( ν ) . (3.33)Plugging (3.33) into (3.32), we getlim sup n → + ∞ n log P n { L n / ∈ K L } ≤ − L, (3.34)which completes the proof. (cid:3) Acknowledgements
The authors are grateful to the referees for their careful reading of the paper and formany valuable comments and suggestions.
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Wei Liu, School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei430072, PR China; Computational Science Hubei Key Laboratory, Wuhan University,Wuhan, Hubei 430072, PR China.
E-mail address : [email protected] Liming Wu, Laboratoire de Math´ematiques Blaise Pascal, CNRS-UMR 6620, Univer-sit´e Clermont-Auvergne, 3 Place Vasarely, 63178 Aubi`ere, France.
E-mail address ::