Central Limit Results for Jump-Diffusions with Mean Field Interaction and a Common Factor
aa r X i v : . [ m a t h . P R ] S e p Central Limit Results for Jump-Diffusions withMean Field Interaction and a Common Factor.
Amarjit Budhiraja ∗ , Elisabeti Kira † and Subhamay Saha ‡ September 16, 2018
Abstract:
A system of N weakly interacting particles whose dynamics is given in terms of jump-diffusions with a common factor is considered. The common factor is described through anotherjump-diffusion and the coefficients of the evolution equation for each particle depend, in additionto its own state value, on the empirical measure of the states of the N particles and the commonfactor. A Central Limit Theorem, as N → ∞ , is established. The limit law is described in termsof a certain Gaussian mixture. An application to models in Mathematical Finance of self-excitedcorrelated defaults is described. AMS 2000 subject classifications:
Keywords:
Mean field interaction, common factor, weakly interacting jump-diffusions, propa-gation of chaos, central limit theorems, fluctuation limits, symmetric statistics, multiple Wienerintegrals, self-excited correlated defaults.
1. Introduction
For N ≥
1, let Z N, , · · · Z N,N be R d valued stochastic processes, representing trajectories of N particles,which are described through stochastic differential equations (SDE) driven by mutually independentBrownian motions(BM) and Poisson random measures(PRM) such that the statistical distribution of( Z N, , . . . , Z N,N ) is exchangeable. The dependence between the N stochastic processes enters throughthe coefficients of the SDE which, for the i -th process, depend in addition to the i -th state process,on a common stochastic process (common factor) and the empirical measure µ Nt = N P Ni =1 δ Z N,it . Thecommon factor is a m -dimensional stochastic process described once more through a SDE driven bya BM and a PRM which are independent of the other noise processes. Such stochastic systems arecommonly referred to as weakly interacting Markov processes and have a long history. Some of theclassical works include McKean[15, 16], Braun and Hepp [1], Dawson [3], Tanaka [23], Oelschal¨ager [19],Sznitman [21, 22], Graham and M´el´eard [8], Shiga and Tanaka [20], M´el´eard [17]. All of these paperstreat the setting where the ‘common factor’ is absent. Most of this research activity is centered aroundproving Law of Large Number results and Central Limit Theorems(CLT). For example one can show(cf. [21, 19]) that under suitable conditions, if the joint initial distributions of every set of k -particles,for every k , converge to product measures as N → ∞ then the same is true for the joint distributionof the stochastic processes(considered as path space valued random variables) as well. Such a result,referred to as the propagation of chaos is one of the key first steps in the study of the fluctuation theoryfor such a system of interacting particles.Systems with a common factor arise in many different areas. In Mathematical Finance, they havebeen used to model correlations between default probabilities of multiple firms[2]. In neuroscience mod-eling these arise as systematic noise in the external current input to a neuronal ensemble[6]. For particle ∗ Research supported in part by the National Science Foundation (DMS-1004418, DMS-1016441, DMS-1305120) andthe Army Research Office (W911NF-10-1-0158, W911NF- 14-1-0331) † Research support in part by CAPES and Fulbright Commission (Proc. 15856/12-7) ‡ Research supported in part by the Indo-US VI-MSS postdoctoral fellowship1 imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018
CLT for Weakly Interacting Particles approximation schemes for stochastic partial differential equations (SPDE), the common factor corre-sponds to the underlying driving noise in the SPDE[13, 14]. The goal of this work is to study a generalfamily of weakly interacting jump-diffusions with a common factor. Our main objective is to establisha suitable Central Limit Theorem. A key point here is that due to the presence of the common fac-tor, the limit of N P Ni =1 δ Z N,i will in general be a random measure. This in particular means that thecentering in the fluctuation theorem will typically be random as well and one expects the limit law forsuch fluctuations to be not Gaussian but rather a ‘Gaussian mixture’. Our main result is Theorem 2.4which provides a CLT under Conditions 2.1, 2.3, 6.2, 7.1, and 7.4. The summands in this CLT can bequite general functionals of the trajectories of the particles with suitable integrability properties. Thekey idea is to first consider a closely related collection of N stochastic processes that, conditionally on acommon factor, are independent and identically distributed. By introducing a suitable Radon-Nikodymderivative one can evaluate the expectations associated with a perturbed form of the original scaledand centered sum in terms of the conditionally i.i.d. collection. The asymptotics of the latter quantityare easier to analyze using, in particular, the classical limit theorems for symmetric statistics[4]. Theperturbation arises due to the fact that in the original system the evolution of the common factor jump-diffusion depends on the empirical measure of the states of the N -particles whereas in the conditionallyi.i.d. construction the common factor evolution is determined by the large particle limit of the empiricalmeasures. Estimating the error introduced by this perturbation is one of the key technical challengesin the proof.In a setting where there is no common factor such central limit results have been obtained in theclassical works of Sznitman[22] and Shiga and Tanaka[20]. In this case the limit law is Gaussian andthe probability law of the actual N -particle system can be realized exactly through a simple absolutelycontinuous change of measure from the probability law of an i.i.d. system. Another aspect that makesthe analysis in the current work significantly more challenging is that unlike [22, 20] the dependence ofthe coefficients of the model on the empirical measure in nonlinear.Central limit theorems for systems of weakly interacting particles with a common factor have previ-ously been studied in [14]. This work is motivated by applications to particle system approximations tosolutions of SPDE. In addition to the fact that the form of the common factor in [14] is quite differentfrom that in our work, there are several differences between these two works. The model consideredin the current work allows for jumps in both particle dynamics and the common factor dynamics nei-ther of which are present in [14]. Also, in [14] the fluctuation limit theorem is established for centeredand scaled empirical measures considered as stochastic processes in the space of (modified) Schwartzdistributions which in practice yields a functional central limit theorem for smooth functionals thatdepend on just the current state of the particles. In contrast, the current work allows for very generalsquare integrable functionals that could possibly depend on the whole trajectory of the particles. Thus,in particular, unlike [14], one can obtain from our work limit theorems for statistics that depend onthe particle states at multiple time instants. In Section 8 we sketch an argument that shows how onecan recover convergence of modified Schwartz distribution valued stochastic processes from our mainconvergence result (Theorem 2.4). A key difference in the argument here (from [14]) is that we do notrequire unique solvability results for SPDE in order to characterize the limit. More precisely, in [14] thelimit law is characterized through the solution of a certain SPDE and one of the key technical challengesis proving the wellposedness of the equation, whereas in the current work the description of the limitlaw is given in terms of a certain mixture of Gaussian distributions (see (2.11)). We note that in somerespects the results in [14] are more general in that they allow for infinite dimensional common factorsand weighted empirical measures. Our proofs rely on a Girsanov change of measure which requires thediffusion coefficients to satisfy a suitable non-degeneracy condition. Although the proofs in [14] are quitedifferent and the form of state dependence allowed there is somewhat more general, it is interesting tonote that the approach taken in [14] also requires a non-degeneracy condition on the diffusion coefficient(see Condition (S4) in Section 4 of [14]).One of our motivations for the current study is to establish central limit results for models in Mathe- imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles matical Finance of self-exciting correlated defaults[2]. In Section 9 we describe how Theorem 2.4 yieldssuch results.The paper is organized as follows. In Section 2 we begin by introducing our model of weakly interact-ing jump-diffusions with a common factor. Next in Section 2.1 we present a basic condition(Condition2.1) that will ensure pathwise existence and uniqueness of solutions to the SDE for the N -particle sys-tem and also for a related family of SDE describing a nonlinear Markov process. These wellposednessresults are given in Theorem 2.2 the proof of which is given in Section 3. The proofs are based onideas and results from [12, 11, 14]. In Section 2.2 we present the main Central Limit Theorem of thiswork. Introducing conditions for the theorem requires some notation and thus we postpone some ofthem to later sections (specifically Conditions 6.2, 7.1 and 7.4 are introduced in Sections 6, 7.1 and7.2 respectively). Section 3 is devoted to the proof of Theorem 2.2. In Section 4 we recall the classi-cal result of Dynkin and Mandelbaum[4] on limit laws of degenerate symmetric statistics described interms of multiple Wiener integrals. Section 5 introduces the Girsanov change of measure that is the keyingredient in our proofs. Section 6 enables the estimation of the error due to the perturbation describedearlier in the Introduction and Section 7 contains the proof of Theorem 2.4. In Section 8, using Theorem2.4, we sketch an argument for proving weak convergence of scaled and centered empirical measures asstochastic processes with values in the dual of a suitable Nuclear space. Finally Section 9 discusses anapplication of Theorem 2.4 to certain models in Mathematical Finance.The following notations will be used. Fix T < ∞ . All stochastic processes will be considered over thetime horizon [0 , T ]. We will use the notations { X t } and { X ( t ) } interchangeably for stochastic processes.Space of probability measures on a Polish space S , equipped with the topology of weak convergence,will be denoted by P ( S ). A convenient metric for this topology is the bounded-Lipschitz metric d BL defined as d BL ( ν , ν ) = sup f ∈ B |h f, ν − ν i| , ν , ν ∈ P ( S ) , where B is the collection of all Lipschitz functions f that are bounded by 1 and such that the corre-sponding Lipschitz constant is bounded by 1 as well; and h f, µ i = R f dµ for a signed measure µ on S and µ -integrable f : S → R . For a function f : [0 , T ] → R k , k f k ∗ ,t . = sup ≤ s ≤ t k f ( s ) k , t ∈ [0 , T ]. Also,for µ i : [0 , T ] → P ( S ), i = 1 , d BL ( µ , µ ) ∗ ,t = sup ≤ s ≤ t d BL ( µ ( s ) , µ ( s )) . Borel σ -field on a Polish space S will be denoted as B ( S ). Space of functions that are right continuous withleft limits (RCLL) from [0 , ∞ ) [resp. [0 , T ]] to S will be denoted as D S [0 , ∞ ) [resp. D S [0 , T ]] and equippedwith the usual Skorohod topology. Similarly C S [0 , ∞ ) [resp. C S [0 , T ]] will be the space of continuousfunctions from [0 , ∞ ) [resp. [0 , T ]] to S , equipped with the local uniform [resp. uniform] topology. For x ∈ D S [0 , T ] and t ∈ [0 , T ], x [0 ,t ] will denote the element of D S [0 , t ] defined as x [0 ,t ] ( s ) = x ( s ), s ∈ [0 , t ].Also given x [0 ,t ] ∈ D S [0 , t ], x [0 ,t ] ( s ) will be written as x s . Similar notation will be used for stochasticprocesses.For a bounded function f from S to R , k f k ∞ = sup x ∈ S | f ( x ) | . Probability law of a S valued randomvariable η will be denoted as L ( η ) and its conditional distribution (a P ( S ) valued random variable)given a sub- σ field G will be denoted as L ( η | G ). Convergence of a sequence { X n } of S valued randomvariables in distribution to X will be written as X n ⇒ X . For a σ -finite measure ν on a Polish space S , L R k ( S , ν ) will denote the Hilbert space of ν -square integrable functions from S to R k . When k = 1, wewill merely write L ( S , ν ). The norm in this Hilbert space will be denoted as k · k L ( S ,ν ) . We will usuallydenote by κ, κ , κ , · · · , the constants that appear in various estimates within a proof. The values ofthese constants may change from one proof to another. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles
2. Main results
Let X t = [0 , t ] × R d × R + , X t = [0 , t ] × R m × R + , t ∈ [0 , T ] . For k ∈ N , let C k and D k denote C R k [0 , T ] and D R k [0 , T ] respectively. Let M d [resp. M m ] be the spaceof σ -finite measures on X T [resp. X T ] with the topology of vague convergence.For fixed N ≥
1, consider the system of equations for the R d valued RCLL stochastic processes Z N,i , i = 1 , . . . N and the R m valued RCLL process U N given on a filtered probability space ( Ω, F , P , {F t } ): Z N,it = Z N,i + Z t b ( Z N,is , U Ns , µ Ns ) ds + Z t σ ( Z N,is , U Ns , µ Ns ) dB is + Z X t ψ d ( Z N,is − , U Ns − , µ Ns − , u, h ) d n i (2.1) U Nt = U + Z t b ( U Ns , µ Ns ) ds + Z t σ ( U Ns , µ Ns ) dB s + Z X t ψ d ( U Ns − , µ Ns − , u, h ) d n , (2.2)Here B i , i ∈ N are r -dimensional Brownian motions(BM); B is a m dimensional BM; n i , i ∈ N are Poisson random measures (PRM) with intensity measure ν = λ T ⊗ γ ⊗ λ ∞ on X T , where λ T [resp. λ ∞ ] is the Lebesgue measure on [0 , T ] [resp. [0 , ∞ )] and γ is a finite measure on R d ; n is aPRM with intensity measure ν = λ T ⊗ γ ⊗ λ ∞ on X T , where γ is a finite measure on R m . Allthese processes are mutually independent and they have independent increments with respect to thefiltration {F t } . Also µ Ns = N P Ni =1 δ Z N,is and ψ d , ψ d are maps defined as follows: For ( x, y, θ, u, h, k ) ∈ R d × R m × P ( R d ) × R + × R d × R m ψ d ( x, y, θ, u, h ) = h [0 ,d ( x,y,θ,h )] ( u ) , ψ d ( y, θ, u, k ) = k [0 ,d ( y,θ,k )] ( u ) , where d and d are nonnegative maps on R d + m × P ( R d ) × R d and R m × P ( R d ) × R m respectively.Roughly speaking, given ( Z N,it − , U Nt − , µ Nt − ) = ( x, y, θ ), the jump for Z N,i at instant t occurs at rate R R d d ( x, y, θ, h ) γ ( dh ) and the jump distribution is given as c · d ( x, y, θ, h ) γ ( dh ) where c is the normal-ization constant. Jumps of U N are described in an analogous manner.We assume that { Z N,i } Ni =1 are i.i.d. with common distribution µ and U is independent of { Z N,i } Ni =1 and has probability distribution ρ . Also, { Z N,i } Ni =1 and U are F measurable.Conditions on the various coefficients will be introduced shortly. Along with the N -particle equations(2.1)-(2.2) we will also consider a related infinite system of equations for R d × R m valued RCLL stochasticprocesses ( X i , Y ), i ∈ N given on ( Ω, F , P , {F t } ). X it = X i + Z t b ( X is , Y s , µ s ) ds + Z t σ ( X is , Y s , µ s ) dB is + Z X t ψ d ( X is − , Y s − , µ s − , u, h ) d n i (2.3) Y t = Y + Z t b ( Y s , µ s ) ds + Z t σ ( Y s , µ s ) dB s + Z X t ψ d ( Y s − , µ s − , u, h ) d n , (2.4)Here µ t = lim k →∞ k P ki =1 δ X it , where the limit is a.s. in P ( R d ). As for the N -particle system, we assumethat { X i } i ∈ N are i.i.d. with common distribution µ and Y is independent of X ≡ { X i } i ∈ N and hasprobability distribution ρ . Also, { X i } i ∈ N and Y are F measurable. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles We now give conditions on the coefficient functions under which the systems of equations (2.1)-(2.2)and (2.3)-(2.4) have unique pathwise solutions. A pathwise solution of (2.3)-(2.4) is a collection ofRCLL processes ( X i , Y ), i ≥
1, with values in R d × R m such that: (a) Y is {G t } adapted, where G s = σ { Y , B r , n ([0 , r ] × A ) , r ≤ s, A ∈ B ( R m × R + ) } ; (b) X is {F t } adapted where X = ( X i ) i ∈ N ;(c) stochastic integrals on the right sides of (2.3)-(2.4) are well defined; (d) Equations (2.3)-(2.4) holda.s. Uniqueness of pathwise solutions says that if ( X, Y ) and ( X ′ , Y ′ ) are two such solutions with( X , Y ) = ( X ′ , Y ′ ) then they must be indistinguishable. Existence and uniqueness of solutions to(2.1)-(2.2) are defined in a similar manner. In particular, in this case (a) and (b) are replaced by therequirement that ( Z N,i , U N ) Ni =1 are {F t } adapted.We now introduce conditions on the coefficients that will ensure existence and uniqueness of solutions. Condition 2.1.
There exist ǫ, K ∈ (0 , ∞ ) such that(a) For all z = ( x, y ) ∈ R d × R m , ν ∈ P ( R d ) , ( h, k ) ∈ R d × R m , ǫ ≤ d ( z, ν, h ) ≤ K, ≤ d ( y, ν, k ) < K, Z R d k h k γ ( dh ) ≤ K , Z R m k k k γ ( dk ) ≤ K , and max {k σ ( z, ν ) k , k σ ( y, ν ) k , k b ( z, ν ) k , k b ( y, ν ) k} ≤ K. (b) For all z = ( x, y ) , z ′ = ( x ′ , y ′ ) ∈ R d × R m , ν, ν ′ ∈ P ( R d ) the functions σ, σ , b, b satisfy k σ ( z, ν ) − σ ( z ′ , ν ′ ) k + k σ ( y, ν ) − σ ( y ′ , ν ′ ) k ≤ K ( k z − z ′ k + d BL ( ν, ν ′ )) k b ( z, ν ) − b ( z ′ , ν ′ ) k + k b ( y, ν ) − b ( y ′ , ν ′ ) k ≤ K ( k z − z ′ k + d BL ( ν, ν ′ )) and the functions d, d satisfy Z R d k h k k d ( z, ν, h ) − d ( z ′ , ν ′ , h ) k γ ( dh ) ≤ K ( k z − z ′ k + d BL ( ν, ν ′ )) Z R m k k k k d ( y, ν, k ) − d ( y ′ , ν ′ , k ) k γ ( dk ) ≤ K ( k y − y ′ k + d BL ( ν, ν ′ ))Under the above condition we can establish the following wellposedness result. Theorem 2.2.
Suppose that Z k x k µ ( dx ) + Z k y k ρ ( dy ) < ∞ (2.5) and Condition 2.1 holds. Then:(a) the system of equations (2.3) - (2.4) has a unique pathwise solution.(b) the system of equations (2.1) - (2.2) has a unique pathwise solution. Proof of the theorem is given in Section 3.
Remark 2.1. (i) We note that the unique pathwise solvability in (a) implies that there is a measurablemap U : R m × C m × M m → D m such that the solution Y of (2.4) is given as Y = U ( Y , B , n ).(ii) Recall that G s = σ { Y , B r , n ([0 , r ] × A ) , r ≤ s, A ∈ B ( R m × R + ) } , s ∈ [0 , T ]. Let G = G T . Thenexactly along the lines of Theorem 2.3 of [13] it follows that if ( { X i } , Y ) is a solution of (2.3)-(2.4) then µ t = L ( X i ( t ) | G ) = L ( X i ( t ) | G t ) , t ∈ [0 , T ] , i ∈ N . (2.6)In particular, there is a measurable map Π : R m ×C m ×M m → D P ( R d ) [0 , T ] such that Π( Y , B , n ) = µ a.s. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles The main result of this work establishes a CLT for N P Ni =1 δ Z N,i . For that, we will make additionalassumptions on the coefficients.
Condition 2.3.
For some p ∈ N , r = d + p and for all ( x, y, ν, k ) ∈ R d × R m × P ( R d ) × R m , σ ( x, y, ν ) = [ I d × d , ˜ σ ( x )] , d ( y, ν, k ) = d ( k ) where I d × d is the d × d identity matrix and ˜ σ ( x ) is a d × p matrix. Note that σ is allowed to depend on ( y, ν ). Remark 2.2.
It is easily seen that if b is of the form b = (˜ b, ′ where for some q < d , ˜ b is a R q valuedfunction then one can relax the assumption on σ by allowing it to be of the form σ ( x, y, ν ) = (cid:18) I q × q ˜ σ ( x )˜ σ ( x ) ˜ σ ( x ) (cid:19) . We will need additional smoothness assumptions on the coefficients b, d, b and σ (Conditions 6.2,7.1 and 7.4) however stating them requires some notation which we prefer to introduce in later sections.As argued in Section 7.3, these conditions are satisfied quite generally. Below is the main result of thiswork. We begin by introducing the following canonical spaces and stochastic processes. LetΩ d = C r × M d × D d , Ω m = C m × M m × D m , Recall from (2.3)-(2.4) the processes ( B i , n i ) i ∈ N and the pathwise solution ( { X i } i ∈ N , Y ). Define for N ∈ N the probability measure P N on ¯ Ω N = Ω m × Ω Nd as P N = L (cid:0) ( B , n , Y ) , ( B , n , X ) , . . . , ( B N , n N , X N ) (cid:1) Note that P N can be disintegrated as P N ( dω dω · · · dω N ) = α ( ω , dω ) · · · α ( ω , dω N ) P ( dω ) , (2.7)where P = L ( B , n , Y ). For ¯ ω = ( ω , ω , . . . , ω N ) ∈ ¯ Ω N , V i (¯ ω ) = ω i , i = 0 , , . . . , N and abusingnotation, V i = ( B i , n i , X i ) , i = 1 , . . . , N, V = ( B , n , Y ) . (2.8)Also define the canonical process V ∗ = ( B ∗ , n ∗ , X ∗ ) on Ω d as V ∗ ( w ) = ( B ∗ ( w ) , n ∗ ( w ) , X ∗ ( w )) = ( w , w , w ); w = ( w , w , w ) ∈ Ω d . (2.9)We denote by A the collection of all measurable maps ϕ : D d → R such that ϕ ( X ∗ ) ∈ L ( Ω d , α ( ω , · ))for P a.e. ω ∈ Ω m . For ϕ ∈ A and ω ∈ Ω m , let m ϕ ( ω ) = Z Ω d ϕ ( X ∗ ( ω )) α ( ω , dω ) , Φ ω = ϕ ( X ∗ ) − m ϕ ( ω ) . (2.10)Let for ω ∈ Ω m and ϕ ∈ A , σ ϕω ∈ R + be defined through (7.42). Denote by π ϕω the normal distributionwith mean 0 and standard deviation σ ϕω . Let π ϕ ∈ P ( R ) be defined as π ϕ = Z Ω m π ϕω P ( dω ) . (2.11)Finally with { Z N,i } Ni =1 as defined in (2.1) and ϕ ∈ A , let¯ V ϕN = √ N N N X j =1 ϕ ( Z N,j ) − m ϕ ( ¯ V ) , where ¯ V = ( B , n , U ( U , B , n )) and U is as introduced below Theorem 2.2. Denote by π ϕN ∈ P ( R )the probability distribution of ¯ V ϕN . The following is the main result of this work. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Theorem 2.4.
Suppose that Conditions 2.1, 2.3, 6.2, 7.1 and 7.4 hold. Then, for all ϕ ∈ A , π ϕN converges weakly to π ϕ as N → ∞ . Rest of the paper is organized as follows. In Section 3 we present the proof of the wellposednessresult (Theorem 2.2). Section 4 recalls some classical results of Dynkin and Mandelbaum[4] on limitsof symmetric statistics. In Section 5 we introduce the Girsanov change of measure that plays a key rolein proofs and Section 6 gives some moment bounds that will be frequently appealed to in our proofs.Section 7 contains the proof of our main result (Theorem 2.4). In Section 8 we discuss how Theorem 2.4can be used to prove central limit theorems for centered and scaled empirical measures. Finally Section9 considers an application of our results to certain models in mathematical finance.
3. Proof of Theorem 2.2.
Proof of the theorem follows along the lines of [13], we sketch the argument for the first statement inTheorem 2.2 and omit the proof of the second statement. Namely, we show now that if { X i , i ∈ N } and Y are as defined below (2.4); (2.5) holds; and Condition 2.1 is satisfied, then the systems ofequations (2.3)-(2.4) has a unique pathwise solution. We first argue pathwise uniqueness. Suppose that R = { R i = ( X i , Y ) , i ∈ N } and ˜ R = { ˜ R i = ( ˜ X i , ˜ Y ) , i ∈ N } are two solutions of (2.3)-(2.4) with R = ˜ R . Then using Condition 2.1 and standard maximal inequalities, for t ∈ [0 , T ], E (cid:13)(cid:13)(cid:13)(cid:13)Z · [ σ ( R is , µ s ) − σ ( ˜ R is , ˜ µ s )] dB is (cid:13)(cid:13)(cid:13)(cid:13) ∗ ,t ≤ κ K E (cid:20)Z t ( k R i − ˜ R i k ∗ ,s + d BL ( µ · , ˜ µ · ) ∗ ,s ) ds (cid:21) / ≤ κ K √ t E ( k R i − ˜ R i k ∗ ,t + d BL ( µ · , ˜ µ · ) ∗ ,t ) . Here, ˜ µ t = lim k →∞ k P ki =1 δ ˜ X it and κ is a global constant. Similarly, E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z [0 , · ] × R d [ ψ d ( R is − , µ s − , u, h ) − ψ d ( ˜ R is − , ˜ µ s − , u, h )] d n i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∗ ,t ≤ E Z X t k ψ d ( R is − , µ s − , u, h ) − ψ d ( ˜ R is − , ˜ µ s − , u, h ) k d n i ≤ E Z [0 ,t ] × R d k h k| d ( R is , µ s , h ) − d ( ˜ R is , ˜ µ s , h ) | γ ( dh ) ds ≤ κ Z [0 ,t ] E ( k R i − ˜ R i k ∗ ,s + d BL ( µ · , ˜ µ · ) ∗ ,s ) ds, where the last inequality uses Condition 2.1(b). One has analogous estimates for terms involving σ , d , b and b . Also by Fatou’s lemma, E d BL ( µ · , ˜ µ · ) ∗ ,s = E sup ≤ u ≤ s sup f ∈ B |h f, µ u − ˜ µ u i| ≤ lim inf k →∞ k k X i =1 E k X i − ˜ X i k ∗ ,s ≤ sup i E k X i − ˜ X i k ∗ ,s . Letting a t = sup i E k X i − ˜ X i k ∗ ,t + E k Y − ˜ Y k ∗ ,t , t ∈ [0 , T ]we then have from the above estimates that for some κ ∈ (0 , ∞ ) a t ≤ κ ( Z t a s ds + √ ta t ) , t ∈ [0 , T ] imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Taking t sufficiently small we see now that a s = 0 for all s ∈ [0 , t ]. A recursive argument then showsthat a s = 0 for all s ∈ [0 , T ]. This completes the proof of uniqueness.Next we prove existence of solutions. We will use ideas and results from [13] (specifically Lemma 2.1and Theorem 2.2 therein). Define for t ∈ [0 , T ] and n ≥ D n ( t ) = ⌊ nt ⌋ n , B n,it = B i ⌊ nt ⌋ n , B n, t = B ⌊ nt ⌋ n , n n,i ( A × [0 , t ]) = n i ( A × [0 , ⌊ nt ⌋ n ]), n n, ( A × [0 , t ]) = n ( A × [0 , ⌊ nt ⌋ n ]), A ∈ B ( R d × [0 , T ]), A ∈B ( R m × [0 , T ]). Let R n . = ( R n,i = ( X n,i , Y n ) , i ∈ N ) be the solution of (2.3)-(2.4) with dt , ( B i , B , n i , n )and µ t replaced by dD n ( t ), ( B n,i , B n, , n n,i , n n, ) and µ nt = lim k →∞ k P ki =1 δ X n,it , respectively. Notethat the solution is determined recursively over intervals of length 1 /n and µ nt is well defined for every t ∈ [0 , T ] since lim k →∞ k P ki =1 δ X n,it exists a.s. from the exchangeability of { X n,it , i ∈ N } which in turnis a consequence of the exchangeability of { X i , i ∈ N } . Using the boundedness of the coefficients it isstraightforward to check that E (cid:16) k R n,it + r − R n,it k | F t (cid:17) ≤ κ ⌊ n ( t + r ) ⌋ − ⌊ nt ⌋ n , t ∈ [0 , T − r ] , r ≥ , i ∈ N , where κ is a constant independent of n, i, t, r . It then follows that for each i ∈ N , { R n,i } n ∈ N is tight in D R d × R m [0 , T ]. This proves tightness of the sequence { R n } n ∈ N in ( D R d × R m [0 , T ]) ⊗∞ . A similar estimateas in the above display shows that for i, j ∈ N , { R n,i + R n,j } n ∈ N is tight in D R d × R m [0 , T ]. Thus wehave that { R n } n ∈ N is tight in D ( R d × R m ) ⊗∞ [0 , T ] (see for example [5], Problems 3.11.22 and 3.11.23).Let ¯ R . = { ¯ R i = ( ¯ X i , ¯ Y ) } i ∈ N denote a sub-sequential weak limit point. Then { ¯ X i } is exchangeable aswell and so ¯ µ t = lim k →∞ k P ki =1 δ ¯ X it is well defined where the limit exists a.s. From Lemma 2.1 in [13](see also [11]) it now follows that (along the chosen subsequence) ( R n , µ n ) converges in distribution to( ¯ R, ¯ µ ), in D ( R d × R m ) ⊗∞ ×P ( R d ) [0 , T ].We note that ψ d regarded as a map from R d × R m × P ( R d ) to L R d ( R + × R d , λ ∞ ⊗ γ ) is a continuousmap. Indeed for z = ( x, y ), z ′ = ( x ′ , y ′ ) ∈ R d × R m and ν, ν ′ ∈ P ( R d ) Z R + × R d k ψ d ( z, ν, u, h ) − ψ d ( z ′ , ν ′ , u, h ) k duγ ( dh ) = Z R d k h k | d ( z, ν, h ) − d ( z ′ , ν ′ , h ) | γ ( dh ) ≤ K ( k z − z ′ k + d BL ( ν, ν ′ ))where the last inequality is from Condition 2.1. Similarly ψ d is a continuous map from R m × P ( R d )to L R m ( R + × R m , λ ∞ ⊗ γ ). Fix p ∈ N , ϕ , · · · ϕ p ∈ L R d ( R + × R d , λ ∞ ⊗ γ ) and ˜ ϕ , · · · ˜ ϕ p ∈ L R m ( R + × R m , λ ∞ ⊗ γ ). Let I n,iϕ j ( t ) = R X t ϕ j ( u, h ) d n n,i , I n,i ˜ ϕ j ( t ) = R X t ˜ ϕ j ( u, k ) d n n, , j = 1 , . . . p , t ∈ [0 , T ].Fix ℓ ∈ N . Consider the vector of processes consisting of σ ( X n,i · , Y n · , µ n · ), b ( X n,i · , Y n · , µ n · ), σ ( Y n · , µ n · ), b ( Y n · , µ n · ), B n,i · , B n, · , I n,iϕ j , I n,i ˜ ϕ j , ψ d ( X n,i · , Y n · , · ), ψ d ( Y n · , · ), i ≤ ℓ , j ≤ p . Then by the continuity of b, b , σ, σ and the continuity property of ψ d , ψ d noted above this vector of processes converges in distri-bution in D E [0 , T ] to the vector of processes obtained by replacing ( X n,i , Y n , µ n , B n,i , B n, , n n,i , n n, )with ( ¯ X i , ¯ Y , ¯ µ, ¯ B i , ¯ B , ¯ n i , ¯ n ). Here E = R k × L R d ( R + × R d , λ ∞ × γ ) × L R m ( R + × R m , λ ∞ × γ ) for asuitable value of k . From Theorem 4.2 of [12] it now follows that ( ¯ X i , ¯ Y ) is a solution of (2.3)-(2.4) with( B i , B , n i , n ) replaced with ( ¯ B i , ¯ B , ¯ n i , ¯ n ) proving the existence of a weak solution of (2.3)-(2.4).From pathwise uniqueness established earlier it now follows that there exists a strong solution of (2.3)-(2.4). Exactly along the lines of the proof of Theorem 2.3 of [13] it follows that { µ t } is {G t } adapted.Also, using Condition 2.1, if ( Y, µ ) and ( ˜
Y , µ ) solve (2.4) then Y and ˜ Y are indistinguishable. From thisand the classical Yamada-Watanabe argument (cf. [10], Theorem IV.1.1) it follows that { Y t } is {G t } adapted as well. This completes the proof of pathwise existence and uniqueness of solutions.
4. Asymptotics of Symmetric Statistics.
The proof of the central limit theorem crucially relies on certain classical results from [4] on limit lawsof degenerate symmetric statistics. In this section we briefly review these results. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018
CLT for Weakly Interacting Particles Let X be a Polish space and let { X n } ∞ n =1 be a sequence of independent identically distributed X -valued random variables having common probability law ν . For k = 1 , , . . . let L ( ν ⊗ k ) be the spaceof all real valued square integrable functions on ( X k , B ( X ) ⊗ k , ν ⊗ k ). Denote by L sym ( ν ⊗ k ) the subspaceof symmetric functions, namely functions φ ∈ L ( ν ⊗ k ) such that for every permutation π on { , · · · k } , φ ( x , · · · , x k ) = φ ( x π (1) , · · · , x π ( k ) ) , ν ⊗ k a.e ( x , . . . x k ) . Given φ k ∈ L sym ( ν ⊗ k ) define a symmetric statistic σ nk ( φ k ) as σ nk ( φ k ) = X ≤ i i
5. Girsanov Change of Measure
For N ∈ N , let ¯ Ω N , P N , V i , i = 0 , . . . , N , Y , µ N be as in Section 2.2. Also let µ = Π( Y , B , n ). Withthese definitions (2.3)-(2.4) are satisfied for i = 1 , . . . , N ; µ s = L ( X i ( s ) | G ) = L ( X i ( s ) | G s ), s ∈ [0 , T ], i = 1 , . . . , N ; and Y is {G t } adapted, where G s = σ { Y , B r , n ([0 , r ] × A ) , r ≤ s, A ∈ B ( R m × R + ) } and G = G T .In addition to the above processes, define Y N as the unique solution of the following equation Y Nt = Y + Z t b ( Y Ns , µ Ns ) ds + Z t σ ( Y Ns , µ Ns ) dB s + Z X t k [0 ,d ( k )] ( u ) d n , (5.1)where µ Ns = N P Ni =1 δ X is .Let for i = 1 , . . . , N , u ∈ R + , h ∈ R d and s ∈ [0 , T ] R i = ( X i , Y ) , R N,i = ( X i , Y N ) , β N,is = b ( R N,is , µ Ns ) − b ( R is , µ s ) , d N,is ( h ) = d ( R N,is , µ Ns , h ) , d is ( h ) = d ( R is , µ s , h ) ,e N,is ( h ) = d N,is ( h ) − d is ( h ) , r N,is ( u, h ) = 1 [0 , d is ( h )] ( u ) log d N,is ( h ) d is ( h ) . Write B i = ( W i , ˜ W i ), where W i , ˜ W i are independent d and p dimensional Brownian motions respec-tively. Define { H N ( t ) } as H N ( t ) = exp (cid:0) J N, ( t ) + J N, ( t ) (cid:1) where J N, ( t ) = N X i =1 (cid:18)Z t β N,is · dW is − Z t k β N,is k ds (cid:19) and J N, ( t ) = N X i =1 Z X t r N,is − ( u, h ) d n i − Z [0 ,t ] × R d e N,is ( h ) γ ( dh ) ds ! . Letting for t ∈ [0 , T ], ¯ F Nt = σ { V i ( s ) , ≤ s ≤ t, i = 0 , . . . , N } , we see that { H Nt } is a ¯ F Nt martingaleunder P N . Define a new probability measure Q N on ¯Ω N by dQ N dP N = H N ( T ) . Expected values under P N and Q N will be denoted as E P N and E Q N respectively.By Girsanov’s theorem, { ( X , . . . , X N , Y N , V ) } has the same probability law under Q N as { ( Z N, , . . . ,Z N,N , U N , ¯ V ) } (defined in (2.1) - (2.2) and above Theorem 2.4) under P N . Thus in order to prove thetheorem it suffices to show thatlim N →∞ E Q N exp (cid:18) i (cid:8) √ N (cid:0) N N X j =1 ϕ ( X j ) − m ϕ ( V ) (cid:1)(cid:9)(cid:19) = Z Ω m exp (cid:18) −
12 ( σ ϕω ) (cid:19) P ( dω ) , imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles which is equivalent to showinglim N →∞ E P N exp (cid:18) i (cid:8) √ N (cid:0) N N X j =1 ϕ ( X j ) − m ϕ ( V ) (cid:1)(cid:9) + J N, ( T ) + J N, ( T ) (cid:19) = Z Ω m exp (cid:18) −
12 ( σ ϕω ) (cid:19) P ( dω ) . (5.2)This will be shown in Section 7.6. We begin with some estimates.
6. Estimating Y N − Y . The following lemma is immediate from the fact that, under P N , { X j } j ∈ N are iid, conditionally on G .We omit the proof. Lemma 6.1.
For each l ∈ N , there exists ϑ l ∈ (0 , ∞ ) such that for all t ∈ [0 , T ]sup k f k ∞ ≤ E P N |h f, µ t − µ Nt i| l ≤ ϑ l N l/ . We now introduce a condition on the coefficients b and σ . Write σ = ( σ , · · · , σ m ), where each σ i is a function with values in R m . Denote by ˆ J the collection of all real functions f on R m + d thatare bounded by 1 and are such that x f (˜ y, x ) is continuous for all ˜ y ∈ R m . We say a function ψ : R m × P ( R d ) → R m is in class S if there exist ˆ c ψ ∈ (0 , ∞ ), a finite subset ˆ J ψ of ˆ J , continuous andbounded functions ψ (1) , ψ (2) from R m × P ( R d ) to R m × m and R m × P ( R d ) × R d to R m respectively; and θ ψ : R m × R m × P ( R d ) × P ( R d ) → R m such that for all y, y ′ ∈ R m and ν, ν ′ ∈ P ( R d ) ψ ( y ′ , ν ′ ) − ψ ( y, ν ) = ψ (1) ( y, ν )( y ′ − y ) + h ψ (2) ( y, ν, · ) , ( ν ′ − ν ) i + θ ψ ( y, y ′ , ν, ν ′ ) , (6.1)where k θ ψ ( y, y ′ , ν, ν ′ ) k ≤ c ψ k y ′ − y k + max f ∈ ˆ J ψ |h f ( y, · ) , ( ν ′ − ν ) i| ! , (6.2)Furthermore, k ψ ( y, ν ) − ψ ( y ′ , ν ′ ) k ≤ c ψ k y − y ′ k + max f ∈ ˆ J ψ |h f ( y, · ) , ( ν − ν ′ ) i| ! . (6.3) Condition 6.2.
The functions b and σ i , i = 1 , · · · , m are in S . Lemma 6.3.
Suppose that Conditions 2.1, 2.3 and 6.2 hold. Then for each l ∈ N , there exists a ˜ ϑ l ∈ (0 , ∞ ) , such that for all t ∈ [0 , T ] E P N k Y Nt − Y t k l ≤ ˜ ϑ l N l/ . Proof.
Fix l ∈ N and t ∈ [0 , T ]. By standard martingale inequalities and property (6.3) for ψ = b , σ i , i = 1 , · · · , m , we have that for some k l ∈ (0 , ∞ ) E P N k Y Nt − Y t k l ≤ k l E P N Z t k Y Ns − Y s k l ds + k l E P N Z t max f ∈ ˆ J |h f ( Y s , · ) , ( µ Ns − µ s ) i| l ds, where ˆ J = ( ˆ J b ) ∪ ( ∪ li =1 ˆ J σ i ). The result is now immediate from Gronwall’s lemma and Lemma 6.1.The following lemma follows on using classical existence/uniqueness results for SDE and an applica-tion of Ito’s formula. We will use the following notation V = ( Y, B , µ ) , Z = ( Y, µ ) , Z N = ( Y N , µ N ) . (6.4) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Lemma 6.4.
Suppose that Conditions 2.1, 2.3 and 6.2 hold. For t ∈ [0 , T ] Y Nt − Y t = 1 N N X j =1 s ,t ( X j [0 ,t ] , V [0 ,t ] ) + T N ( t ) , where T N ( t ) = E t Z t E − s θ b ( Z s , Z Ns ) ds + m X k =1 E t Z t E − s θ σ k ( Z s , Z Ns ) dB ,ks − m X k =1 E t Z t E − s σ k, (1)0 ( Z s ) θ σ k ( Z s , Z Ns ) ds, s ,t ( X j [0 ,t ] , V [0 ,t ] ) = E t Z t E − s b (2) ,c ( Z s , X js ) ds + m X k =1 E t Z t E − s σ k, (2) ,c ( Z s , X js ) dB ,ks − m X k =1 E t Z t E − s σ k, (1)0 ( Z s ) σ k, (2) ,c ( Z s , X js ) ds, {E t } solves the m × m dimensional SDE E t = I + Z t b (1)0 ( Z s ) E s ds + m X k =1 Z t σ k, (1)0 ( Z s ) E s dW ks ,b (2) ,c ( y, ν, ˜ x ) = b (2)0 ( y, ν, ˜ x ) − Z R d b (2)0 ( y, ν, x ′ ) ν ( dx ′ ) , ( y, ν, ˜ x ) ∈ R m × P ( R d ) × R d and σ k, (2) ,c is defined similarly. Proof.
Using (6.1) with ψ = b , σ i , i = 1 , · · · , m , we have that Y Nt − Y t = Z t ( b ( Z Ns ) − b ( Z s )) ds + Z t ( σ ( Z Ns ) − σ ( Z s )) dB ( s )= Z t (cid:16) b (1)0 ( Z s )( Y Ns − Y s ) + h b (2)0 ( Z s , · ) , µ Ns − µ s i + θ b ( Z s , Z Ns ) (cid:17) ds + m X k =1 Z t (cid:16) σ k, (1)0 ( Z s )( Y Ns − Y s ) + h σ k, (2)0 ( Z s , · ) , µ Ns − µ s i + θ σ k ( Z s , Z Ns ) (cid:17) dB ,ks . A standard application of Ito’s formula shows Y Nt − Y t = E t Z t E − s (cid:16) h b (2)0 ( Z s , · ) , µ Ns − µ s i + θ b ( Z s , Z Ns ) (cid:17) ds + m X k =1 E t Z t E − s (cid:16) h σ k, (2)0 ( Z s , · ) , µ Ns − µ s i + θ σ k ( Z s , Z Ns ) (cid:17) dB ,ks − m X k =1 E t Z t E − s σ k, (1)0 ( Z s ) (cid:16) h σ k, (2)0 ( Z s , · ) , µ Ns − µ s i + θ σ k ( Z s , Z Ns ) (cid:17) ds. The result now follows on rearranging terms and noting that h b (2)0 ( Z s , · ) , µ Ns − µ s i = 1 N N X j =1 b (2) ,c ( Z s , X js ) , h σ k, (2)0 ( Z s , · ) , µ Ns − µ s i = 1 N N X j =1 σ k, (2) ,c ( Z s , X js ) . imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles The following lemma follows on using the boundedness of coefficients, an application of Gronwall’slemma, Holder’s inequality, Lemmas 6.1 and 6.3 and properties of θ ψ for ψ in class S . The proof isomitted. Let h jt = s ,t ( X j [0 ,t ] , V [0 ,t ] ), t ∈ [0 , T ]. Lemma 6.5.
Suppose that Conditions 2.1, 2.3 and 6.2 hold. Then for each l ∈ N sup n ∈ N sup t ∈ [0 ,T ] (cid:0) E P N kE t k l + E P N kE − t k l (cid:1) < ∞ and there exists ̟ ∈ (0 , ∞ ) such that for all t ∈ [0 , T ] E P N k N N X j =1 h jt k ≤ ̟N , E P N kT N ( t ) k ≤ ̟N .
7. Proof of Theorem 2.4. J N, . In Lemmas 7.2 and 7.3 below we study the asymptotics of the first and second sums in J N, respectively.For this we introduce an additional condition on the coefficient b . Denote by J the collection of all realfunctions f on R d + m + d that are bounded by 1 and are such that x f (˜ x, ˜ y, x ) is continuous for all(˜ x, ˜ y ) ∈ R d + m . Condition 7.1.
There exist c b ∈ (0 , ∞ ) ; a finite subset J F of J ; continuous and bounded functions b , b from R d + m × P ( R d ) to R d × m and R d + m × P ( R d ) × R d to R d respectively; and θ b : R d + m × R d + m ×P ( R d ) × P ( R d ) → R d such that for all z = ( x, y ) , z ′ = ( x, y ′ ) ∈ R d + m and ν, ν ′ ∈ P ( R d ) b ( z ′ , ν ′ ) − b ( z, ν ) = b ( z, ν )( y ′ − y ) + h b ( z, ν, · ) , ( ν ′ − ν ) i + θ b ( z, z ′ , ν, ν ′ ) and k θ b ( z, z ′ , ν, ν ′ ) k ≤ c b (cid:18) k y ′ − y k + max f ∈J F |h f ( z, · ) , ( ν ′ − ν ) i| (cid:19) . (7.1) Lemma 7.2.
Suppose that Conditions 2.1, 2.3, 6.2, 7.1 hold. For N ∈ N , N X i =1 Z T β N,is dW is = 1 N X i = j Z T b c ( R is , µ s , X js ) dW is + 1 N X i = j Z T b ( R is , µ s ) h js dW is + R N , where R N converges to in probability, where b c ( x, y, ν, ˜ x ) = b ( x, y, ν, ˜ x ) − Z R d b ( x, y, ν, x ′ ) ν ( dx ′ ) , ( x, y, ν, ˜ x ) ∈ R d + m × P ( R d ) × R d . . Proof.
By Condition 7.1 it follows that, for s ∈ [0 , T ], β N,is = b ( R is , µ s )( Y Ns − Y s ) + h b ( R is , µ s , · ) , ( µ Ns − µ s ) i + ζ N,is , (7.2) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where ζ N,is = θ b ( R is , R N,is , µ s , µ Ns ). Next, from (7.1) we have E P N N X i =1 Z T ζ N,is dW is ! = N X i =1 Z T E P N k ζ N,is k ds ≤ κ N X i =1 Z T E P N ( k Y Ns − Y s k ) ds + κ N X i =1 X f ∈J F Z T E P N ( h f ( R is , · ) , ( µ Ns − µ s ) i ) ds. Since f is bounded by 1; J F is a finite collection; and conditionally on G , X i are i.i.d., the second termon the right side using Lemma 6.1 can be bounded by κ /N for some κ ∈ (0 , ∞ ). Also, from Lemma6.3 the first term converges to 0. Combining the above observations we have, as N → ∞ , N X i =1 Z T ζ N,is dW is → , in probability . (7.3)Now consider the second term in (7.2): N X i =1 Z T h b ( R is , µ s , · ) , ( µ Ns − µ s ) i dW is = 1 N N X i =1 Z T b c ( R is , µ s , X is ) dW is + 1 N X i = j Z T b c ( R is , µ s , X js ) dW is . (7.4)Using the boundedness of b it follows that,1 N N X i =1 Z T b c ( R is , µ s , X is ) dW is → , in probability . (7.5)Finally consider the first term in (7.2). From Lemma 6.4, for t ∈ [0 , T ], N X i =1 Z t b ( R is , µ s )( Y Ns − Y s ) dW is = 1 N N X i =1 Z t b ( R is , µ s ) h is dW is + 1 N X i = j Z t b ( R is , µ s ) h js dW is + N X i =1 Z t b ( R is , µ s ) T N ( s ) dW is . The first term on the right side converges to 0 in probability since b , b ( i )0 , σ k, ( i )0 are bounded. Also,using the boundedness of b and Lemma 6.5, the third term converges to 0 in probability. Result nowfollows on combining the above observation with (7.3), (7.4) and (7.5).For the next lemma we will need some notation. Define functions s ,t , s c ,t from R d × D R d + m ×P ( R d ) [0 , t ]to R as follows: For ( x, x (1)[0 ,t ] , x (2)[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] ) ≡ ( x, ζ [0 ,t ] ) ∈ R d × D R d +2 m ×P ( R d ) [0 , t ] s ,t ( x, ζ [0 ,t ] ) = b ( x, y t , ν t ) s ,t ( ζ (1)[0 ,t ] ) · b ( x, y t , ν t ) s ,t ( ζ (2)[0 ,t ] ) s c ,t ( x, ζ [0 ,t ] ) = s ,t ( x, ζ [0 ,t ] ) − m ,t ( ζ [0 ,t ] ) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where ζ ( i )[0 ,t ] = ( x ( i )[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] ), and the function m ,t from D R d +2 m ×P ( R d ) [0 , t ] to R is defined as m ,t ( ζ [0 ,t ] ) = Z R d s ,t ( x ′ , ζ [0 ,t ] ) ν t ( dx ′ ) . Next, define for t ∈ [0 , T ], functions s ,t , s c ,t from R d × D R d +2 m ×P ( R d ) [0 , t ] to R as follows: s ,t ( x, ζ [0 ,t ] ) = 2 b ( x, y t , ν t ) s ,t ( ζ (1)[0 ,t ] ) · b c ( x, y t , ν t , x (2) t ) s c ,t ( x, ζ [0 ,t ] ) = s ,t ( x, ζ [0 ,t ] ) − m ,t ( ζ [0 ,t ] )where the function m ,t from D R d +2 m ×P ( R d ) [0 , t ] to R is defined as m ,t ( ζ [0 ,t ] ) = 12 X i,j ∈{ , } ,i = j Z R d s ,t ( x ′ , x ( i )[0 ,t ] , x ( j )[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] ) ν t ( dx ′ ) . Also, define functions s , s c from R d + m × P ( R d ) to R as follows: For ( x, x (1) , x (2) , y, ν ) ∈ R d + m ×P ( R d ) s ( x, x (1) , x (2) , y, ν ) = b c ( x, y, ν, x (1) ) · b c ( x, y, ν, x (2) ) , s c ( x, x (1) , x (2) , y, ν ) = s ( x, x (1) , x (2) , y, ν ) − m ( x (1) , x (2) , y, ν ) , where m from R d + m × P ( R d ) to R is defined as m ( x (1) , x (2) , y, ν ) = Z s ( x ′ , x (1) , x (2) , y, ν ) ν ( dx ′ ) . Finally, define m t from D R d +2 m ×P ( R d ) [0 , t ] to R as follows. m t ( ζ [0 ,t ] ) = X i =1 m i,t ( ζ [0 ,t ] ) + m ( x (1) t , x (2) t , y t , ν t ) . Recall the process V from (6.4). Lemma 7.3.
For N ∈ N , N X i =1 Z T k β N,is k ds = 1 N X j = k Z T m t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt + 1 N N X j =1 Z T m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] ) dt + R N , (7.6) where R N converges to in probability. Proof.
For N ∈ N , i = 1 , . . . N and s ∈ [0 , T ] k β N,is k = k b ( R N,is , µ Ns ) − b ( R is , µ s ) k = k b ( R is , µ s )( Y Ns − Y s ) + h b ( R is , µ s , · ) , ( µ Ns − µ s ) i + θ b ( R is , R N,is , µ s , µ Ns ) k = k b ( R is , µ s )( Y Ns − Y s ) k + kh b ( R is , µ s , · ) , ( µ Ns − µ s ) ik + k θ b ( R is , R N,is , µ s , µ Ns ) k + 2 b ( R is , µ s )( Y Ns − Y s ) · h b ( R is , µ s , · ) , ( µ Ns − µ s ) i + T N,i ( s ) , (7.7) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where the term T N,i ( s ) consists of the remaining two crossproduct terms. Using (7.1) and Lemmas 6.1and 6.3, we see that N X i =1 Z T k θ b ( R is , R N,is , µ s , µ Ns ) k ds → N → ∞ . (7.8)Similar estimates show that N X i =1 Z T |T N,i ( s ) | ds → N → ∞ . (7.9)Next, using Lemma 6.4, we have k b ( R is , µ s )( Y Ns − Y s ) k = k N N X j =1 b ( R is , µ s ) h js k + k b ( R is , µ s ) T Ns k + T N,i ( s ) , where T N,i ( s ) is the corresponding crossproduct term. Making use of Lemma 6.5 we can bound E P N , G |T N,i ( s ) | by κ N / for some κ > s ∈ [0 , T ] and i, N ∈ N . Sim-ilarly, the expected value of the second term in the above display can be bounded by κ N for some κ >
0. Thus N X i =1 Z T k b ( R is , µ s )( Y Ns − Y s ) k ds = N X i =1 Z T k N N X j =1 b ( R is , µ s ) h js k ds + ˜ R N , (7.10)where ˜ R N → N → ∞ .Recalling the definition of s ,t Z T N X i =1 k N N X j =1 b ( R is , µ s ) h js k ds = 1 N X i,j,k Z T b ( R is , µ s ) h js · b ( R is , µ s ) h ks ds = 1 N X i,j,k Z T s ,t ( X it , X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt. The above expression can be written as1 N X i,j,k Z T s c ,t ( X it , X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt + 1 N X j = k Z T m ,t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt + 1 N X j Z T m ,t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] ) dt. (7.11)From the boundedness of s c ,t , conditional independence of X i , X j , X k for distinct indices i, j, k and thefact that for all ( x, x (1)[0 ,t ] , x (2)[0 ,t ] , v [0 ,t ] ) ∈ R d × D R d +2 m ×P ( R d ) [0 , t ] E P N s c ,t ( X it , x (1)[0 ,t ] , x (2)[0 ,t ] , v [0 ,t ] ) = E P N s c ,t ( x, X i [0 ,t ] , x (2)[0 ,t ] , v [0 ,t ] )= E P N s c ,t ( x, x (1)[0 ,t ] , X j [0 ,t ] , v [0 ,t ] ) = 0 , it follows that the first term in (7.11) converges to 0 in probability. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Next, kh b ( R is , µ s , · ) , ( µ Ns − µ s ) ik = 1 N X j,k b c ( R is , µ s , X js ) · b c ( R is , µ s , X ks ) . Thus N X i =1 Z T kh b ( R is , µ s , · ) , ( µ Ns − µ s ) ik ds = 1 N X i,j,k Z T b c ( R is , µ s , X js ) · b c ( R is , µ s , X ks ) ds = 1 N X i,j,k Z T s ( X is , X js , X ks , Y s , µ s ) ds The above expression can be rewritten as1 N X i,j,k Z T s c ( X is , X js , X ks , Y s , µ s ) ds + 1 N X j = k Z T m ( X js , X ks , Y s , µ s ) ds + 1 N X j Z T m ( X js , X js , Y s , µ s ) ds. (7.12)As before, the first term in (7.12) converges to 0 in probability.Finally we consider the crossproduct term in (7.7): N X i =1 b ( R is , µ s )( Y Ns − Y s ) · h b ( R is , µ s , · ) , ( µ Ns − µ s ) i = 1 N X i,j,k b ( R is , µ s ) h js · b c ( R is , µ s , X ks )+ 1 N X i,k b ( R is , µ s ) T N ( s ) · b c ( R is , µ s , X ks ) ≡ T N ( s ) + T N ( s )where the equality follows from Lemma 6.4. Using Lemma 6.5 we see that R T T N ( s ) ds converges to 0in probability as N → ∞ . For the term T N ( s )2 N X i,j,k Z T b ( R is , µ s ) h js · b c ( R is , µ s , X ks ) ds = 1 N X i,j,k Z T s c ,t ( X it , X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt + 1 N X j = k Z T m ,t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] ) dt + 1 N X j Z T m ,t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] ) dt. The first term on the right side once more converges to 0 in probability. The result now follows oncombining the above display with (7.7), (7.8), (7.9), (7.10), (7.11) and (7.12). J N, . We now consider the term J N, . Recall the constants ǫ, K from Condition 2.1. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles From Taylor’s expansion, there exists a κ ∈ (0 , ∞ ) such that for all α, β ∈ ( ǫ, K )log αβ = ( αβ − −
12 ( αβ − + ϑ ( α, β )( αβ − , where | ϑ ( α, β ) | ≤ κ . Letting ϑ N,is ( h ) = ϑ ( d N,is ( h ) , d is ( h )), we getlog d N,is ( h ) d is ( h ) = d N,is ( h ) d is ( h ) − ! − d N,is ( h ) d is ( h ) − ! + ϑ N,is ( h ) d N,is ( h ) d is ( h ) − ! . Thus Z X T r N,is − ( u, h ) d n i − Z [0 ,T ] × R d e N,is ( h ) γ ( dh ) ds = Z X T [0 , d is − ( h )] ( u ) d N,is − ( h ) d is − ( h ) − ! d ˜ n i − Z X T [0 , d is − ( h )] ( u ) d N,is − ( h ) d is − ( h ) − ! d n i + Z X T [0 , d is − ( h )] ( u ) ϑ N,is − ( h ) d N,is − ( h ) d is − ( h ) − ! d n i , (7.13)where ˜ n i is the compensated PRM: ˜ n i = n i − ν . In the lemmas below we consider the three terms on theright side of (7.13) separately. We introduce the following condition on the coefficient d . Denote by ˜ J thecollection of all real functions f on R d + m +2 d that are bounded by 1 and are such that x f (˜ x, ˜ y, ˜ h, x )is continuous for all (˜ x, ˜ y, ˜ h ) ∈ R d + m + d . Condition 7.4.
There exist c d ∈ (0 , ∞ ) ; a finite subset ˜ J F of ˜ J ; continuous and bounded real functions d , d from R d + m + d ×P ( R d ) to R m and R d + m + d ×P ( R d ) × R d to R respectively; and θ d : R d + m × R d + m ×P ( R d ) × P ( R d ) × R d → R such that for all z = ( x, y ) , z ′ = ( x, y ′ ) ∈ R d + m , h ∈ R d and ν, ν ′ ∈ P ( R d ) d ( z ′ , ν ′ , h ) − d ( z, ν, h ) = ( y ′ − y ) · d ( z, h, ν ) + h d ( z, h, ν, · ) , ( ν ′ − ν ) i + θ d ( z, z ′ , ν, ν ′ , h ) and | θ d ( z, z ′ , ν, ν ′ ) | ≤ c d (cid:18) k y ′ − y k + max f ∈ ˜ J F |h f ( z, h, · ) , ( ν ′ − ν ) i| (cid:19) . (7.14)Next let d c from R d + m × R d × P ( R d ) × R d to R as d c ( x, y, h, ν, ˜ x ) = d ( x, y, h, ν, ˜ x ) − Z R d d ( x, y, h, ν, x ′ ) ν ( dx ′ ) . Lemma 7.5.
For N ∈ N N X i =1 Z X T [0 , d is − ( h )] ( u ) d N,is − ( h ) d is − ( h ) − ! d ˜ n i = 1 N X i = j Z X T [0 , d is − ( h )] ( u ) 1 d is − ( h ) d c ( R is − , h, µ s − , X js − ) d ˜ n i + 1 N X i = j Z X T [0 , d is − ( h )] ( u ) h js d is − ( h ) d ( R is − , h, µ s − ) d ˜ n i + R N , where R N converges to in probability. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Proof.
From Condition 7.4 d N,is ( h ) − d is ( h ) = ( Y Ns − Y s ) · d ( R is , h, µ s )+ h d ( R is , h, µ s , · ) , ( µ Ns − µ s ) i + θ d ( R is , R N,is , µ s , µ Ns , h ) . (7.15)Since N X i =1 θ d ( R is , R N,is , µ s , µ Ns , h ) ≤ c d N X i =1 (cid:18) k Y s − Y Ns k + max f ∈ ˜ J F |h f ( R is , h, · ) , ( µ Ns − µ s ) i| (cid:19) , (7.16)we have from (7.14), Lemma 6.3 and Lemma 6.1 that, as N → ∞ N X i =1 Z X T [0 , d is − ( h )] ( u ) 1 d is − ( h ) θ d ( R is − , R N,is − , µ s − , µ Ns − , h ) d ˜ n i → . (7.17)Next consider the second term on the right side of (7.15). N X i =1 [0 , d is − ( h )] ( u ) 1 d is − ( h ) h d ( R is − , h, µ s − , · ) , ( µ Ns − − µ s − ) i d ˜ n i = 1 N N X i =1 [0 , d is − ( h )] ( u ) 1 d is − ( h ) d c ( R is − , h, µ s − , X is − ) d ˜ n i + 1 N X i = j [0 , d is − ( h )] ( u ) 1 d is − ( h ) d c ( R is − , h, µ s − , X js − ) d ˜ n i . (7.18)Since { n i } Ni =1 are independent, as N → ∞ ,1 N N X i =1 Z X T [0 , d is − ( h )] ( u ) 1 d is − ( h ) d c ( R is − , h, µ s − , X is − ) d ˜ n i → . (7.19)Finally consider the first term on the right side of (7.15). Using Lemma 6.4( Y Ns − Y s ) · d ( R is , h, µ s ) = 1 N N X j =1 h js · d ( R is , h, µ s )+ T Ns · d ( R is , h, µ s ) ds ≡ N N X j =1 h js · d ( R is , h, µ s ) + ˆ T N,i ( s ) . (7.20)Using Lemma 6.5 we see that, as N → ∞ , N X i =1 Z X T [0 , d is − ( h )] ( u ) 1 d is − ( h ) ˆ T N,i ( s − ) d ˜ n i → N N X i,j =1 Z X T [0 , d is − ( h )] ( u ) h js d is − ( h ) · d ( R is − , h, µ s − ) d ˜ n i = 1 N N X i =1 Z X T [0 , d is − ( h )] ( u ) h js d is − ( h ) · d ( R is − , h, µ s − ) d ˜ n i + 1 N X i = j Z X T [0 , d is − ( h )] ( u ) h js d is − ( h ) · d ( R is − , h, µ s − ) d ˜ n i . (7.22) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles As before, using the independence of { n i } Ni =1 , as N → ∞ ,1 N N X i =1 Z X T [0 , d is − ( h )] ( u ) h js d is − ( h ) · d ( R is − , h, µ s − ) d ˜ n i → t ∈ [0 , T ], function ¯ s ,t from R d × D R d + m ×P ( R d ) [0 , t ] × R d to R as follows: For ( x, x (1)[0 ,t ] , x (2)[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] , h ) ≡ ( x, ζ [0 ,t ] ) ∈ R d × D R d +2 m ×P ( R d ) [0 , t ] × R d ¯ s ,t ( x, ζ [0 ,t ] , h ) = 1 d ( x, y t − , ν t − , h ) Y i =1 s ,t ( ζ ( i )[0 ,t ] ) · d ( x, y t − , h, ν t − ) , where ζ ( i )[0 ,t ] = ( x ( i )[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] ). Also define the function ¯ m ,t from D R d +2 m ×P ( R d ) [0 , t ] × R d to R as ¯ m ,t ( ζ [0 ,t ] , h ) = Z R d ¯ s ,t ( x ′ , ζ [0 ,t ] , h ) ν t ( dx ′ ) . Next, define for t ∈ [0 , T ], function ¯ s ,t from R d × D R d +2 m ×P ( R d ) [0 , t ] × R d to R as follows:¯ s ,t ( x, ζ [0 ,t ] , h ) = 2 d c ( x, y t − , h, ν t − , x (2) t − ) d ( x, y t − , ν t − , h ) s ,t ( ζ (1)[0 ,t ] ) · d ( x, y t − , h, ν t − ) . Also define the function ¯ m ,t from D R d + m ×P ( R d ) [0 , t ] × R d to R as¯ m ,t ( ζ [0 ,t ] , h ) = 12 X i,j ∈{ , } ,i = j Z R d ¯ s ,t ( x ′ , x ( i )[0 ,t ] , x ( j )[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] , h ) ν t ( dx ′ ) . Define function ¯ s from R d + m ×P ( R d ) × R d to R as follows: For ( x, x (1) , x (2) , y, ν, h ) ∈ R d + m ×P ( R d ) × R d ¯ s ( x, x (1) , x (2) , y, ν, h ) = Q i =1 d c ( x, y, h, ν, x ( i ) ) d ( x, y, ν, h )and let ¯ m be the function from R d + m × P ( R d ) × R d to R defined as¯ m ( x (1) , x (2) , y, ν, h ) = Z ¯ s ( x ′ , x (1) , x (2) , y, ν, h ) ν ( dx ′ ) . Finally, define ¯ m t from D R d + m ×P ( R d ) [0 , t ] × R d to R as follows.¯ m t ( ζ [0 ,t ] , h ) = X i =1 ¯ m i,t ( ζ [0 ,t ] , h ) + ¯ m ( x (1) t , x (2) t , y t , ν t , h ) . Recall the process V introduced in (6.4). Lemma 7.6.
For N ∈ N N X i =1 Z X T [0 , d is − ( h )] ( u ) d N,is − ( h ) d is − ( h ) − ! d n i = 1 N X j = k Z [0 ,T ] × R d ¯ m t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt + 1 N N X j =1 Z [0 ,T ] × R d ¯ m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt + R N , (7.23) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where R N converges to in probability as N → ∞ . Proof.
From (7.15)( d N,is ( h ) − d is ( h )) = ( T N,i ( s ) + T N,i ( s ) + T N,i ( s )) = X m =1 ( T N,im ( s )) + 2 X m 2, as N → ∞ , N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) |T N,im ( s − ) ||T N,i ( s − ) | d n i → T N,i ( s )) = ( T N,i ( s ) + T N,i ( s )) = ( T N,i ( s )) + ( T N,i ( s )) + 2 T N,i ( s ) T N,i ( s ) , where T N,i ( s ) = 1 N N X j =1 h js · d ( R is , h, µ s ) , T N,i ( s ) = T Ns · d ( R is , h, µ s ) . As for (7.21), we see that, as N → ∞ , N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( T N,i ( s − )) d n i → E P N s ,t ( X j [0 ,t ] , v [0 ,t ] ) = 0, for all j ∈ N and v [0 ,t ] in D R m ×P ( R d ) [0 , t ];and making use of Lemma 6.5 once more, we see that, as N → ∞ , Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) |T N,i ( s − ) T N,i ( s − ) | d n i → N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( T N,i ( s − )) d n i = 1 N X i,j,k Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( h js · d ( R is − , h, µ s − ))( h ks · d ( R is − , h, µ s − )) d n i imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles The above can be rewritten as1 N X i,j,k (cid:16) Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( h js · d ( R is − , h, µ s − ))( h ks · d ( R is − , h, µ s − )) d n i − Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt (cid:17) + 1 N X j = k Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt + 1 N X j Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt. (7.28)A similar argument as below (7.11) shows that the first term in the above display converges to 0 inprobability, as N → ∞ .Combining (7.26), (7.27) and (7.28) we have that N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( T N,i ( s − )) d n i = 1 N X j = k Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X k [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt + 1 N X j Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt + ˜ R N , (7.29)where ˜ R N converges to 0 in probability as N → ∞ .We now consider the term T N,i ( s ). Writing( T N,i ( s )) = 1 N X j,k d c ( R is , h, µ s , X js ) d c ( R is , h, µ s , X ks )we see N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( T N,i ( s − )) d n i = 1 N X i,j,k Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) d c ( R is − , h, µ s − , X js − ) d c ( R is − , h, µ s − , X ks − ) d n i The above can be rewritten as1 N X i,j,k (cid:16) Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) d c ( R is − , h, µ s − , X js − ) d c ( R is − , h, µ s − , X ks − ) d n i − Z [0 ,T ] × R d ¯ m ( X jt , X kt , V t , h ) γ ( dh ) dt (cid:17) + 1 N X j = k Z [0 ,T ] × R d ¯ m ( X jt , X kt , V t , h ) γ ( dh ) dt + 1 N X j Z [0 ,T ] × R d ¯ m ( X jt , X jt , V t , h ) γ ( dh ) dt. (7.30) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles As before, the first term above converges to 0 in probability, as N → ∞ . Thus N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) ( T N,i ( s − )) d n i = 1 N X j = k Z [0 ,T ] × R d ¯ m ( X jt , X kt , Y t , µ t , h ) γ ( dh ) dt + 1 N X j Z [0 ,T ] × R d ¯ m ( X jt , X jt , Y t , µ t , h ) γ ( dh ) dt + ˜ R N , (7.31)where ˜ R N converges to 0 in probability as N → ∞ .We now consider the term 2 T N,i ( s ) T N,i ( s ).2 T N,i ( s ) T N,i ( s ) = 2( Y Ns − Y s ) · d ( R is , h, µ s ) h d ( R is , h, µ s , · ) , ( µ Ns − µ s ) i = 2 N X j,k h js · d ( R is , h, µ s ) d c ( R is , h, µ s , X ks )+ 2 N X k T Ns · d ( R is , h, µ s ) d c ( R is , h, µ s , X ks ) ≡ T N,i ( s ) + T N,i ( s ) . (7.32)For the term T N,i ( s ) note that, N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) T N,i ( s − ) d n i = 2 N X i,j,k Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) h js · d ( R is − , h, µ s − ) d c ( R is − , h, µ s − , X ks − ) d n i (7.33)As in (7.30) and (7.31), we can now write the above as N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) T N,i ( s − ) d n i = 1 N X j = k Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X k [0 ,t ] , Y [0 ,t ] , µ [0 ,t ] , h ) γ ( dh ) dt + 1 N X j Z [0 ,T ] × R d ¯ m ,t ( X j [0 ,t ] , X j [0 ,t ] , Y [0 ,t ] , µ [0 ,t ] , h ) γ ( dh ) dt + ˜ R N , (7.34)where ˜ R N converges to 0 in probability as N → ∞ . Also, as for (7.27), as N → ∞ , N X i =1 Z X T [0 , d is − ( h )] ( u ) 1( d is − ( h )) |T N,i ( s − ) | d n i → ϑ N,is introduced at the beginning of the subsection. Using very similar estimatesas in the proof of Lemma 7.6, one can establish the following result. We omit the proof. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Lemma 7.7. As N → ∞ , N X i =1 Z X T [0 , d is − ( h )] ( u ) ϑ N,is − ( h ) d N,is − ( h ) d is − ( h ) − ! d n i converges to in probability. Conditions 6.2, 7.1 and 7.4 on b , b and d can be regarded as smoothness conditions. These conditionsare satisfied quite generally. We give two examples to illustrate this. Example 7.1. Let d = m = 1. Let ¯ b : R k +2 → R be bounded Lipschitz and twice continuouslydifferentiable, with bounded derivatives, in the last k + 1 variables. Let ¯ b : R k +1 → R be boundedLipschitz and twice continuously differentiable with bounded derivatives. Similar assumptions on ¯ σ i for i = 1 , · · · , m . Let ¯ d : R k +3 → ( ǫ, ∞ ) be bounded and Lipschitz in the first k + 2 variables, uniformlyin the last variable, where ǫ ∈ (0 , ∞ ). Also suppose that ¯ d is twice continuously differentiable, withbounded derivatives, with respect to the middle k + 1 variables. Now let b , b and d be of the form: For( x, y, ν, h ) ∈ R d + m × P ( R d ) × R d • b ( x, y, ν ) = ¯ b ( x, y, h f , ν i , · · · , h f k , ν i ), • b ( y, ν ) = ¯ b ( y, h f , ν i , · · · , h f k , ν i ), • σ i ( y, ν ) = ¯ σ i ( y, h f , ν i , · · · , h f k , ν i ), • d ( x, y, ν, h ) = ¯ d ( x, y, h f , ν i , · · · , h f k , ν i , h ),where f i are bounded Lipschitz functions. Finally let d : R → ( ǫ, ∞ ) be a bounded function and let γ , γ be probability measures on R with finite second moment. Then it is easy to check that Conditions2.1 and 2.3 is satisfied. For Condition 6.2 observe that by Taylor’s expansion, b ( z ′ , ν ′ ) − b ( z, ν ) = ( y ′ − y )¯ b y ( z, h f , ν i , · · · , h f k , ν i )+ k X i =1 ¯ b u i ( z, h f , ν i , · · · , h f k , ν i ) h f i , ( ν ′ − ν ) i + θ b ( z, z ′ , ν, ν ′ ) , where for some constant K , | θ b ( z, z ′ , ν, ν ′ ) | ≤ K ( | y ′ − y | +max i |h f i , ( ν ′ − ν ) i| ). This verifies Condition7.1. Conditions 6.2, 7.4 can be verified similarly. Example 7.2. Let d = m = 1. Let ˜ b : R → R , ˜ b : R → R , ˜ σ i : R → R , i = 1 , · · · , m bebounded Lipschitz functions. Further suppose that ˜ b is twice continuously differentiable with respectto the second variable with bounded derivatives and ˜ b is also twice continuously differentiable withrespect to the first variable, with bounded derivatives. Similar assumptions on ˜ σ i . Let ˜ d : R → R + bebounded and Lipschitz in the first three variables, uniformly in the last variable. Also suppose that ˜ d is twice continuously differentiable, with bounded derivatives, in the second variable. Let d , γ, γ be asin Example 7.1. Now let b , b and d be of the form: • b ( x, y, ν ) = R ˜ b ( x, y, x ′ ) ν ( dx ′ ), • b ( y, ν ) = R ˜ b ( y, x ′ ) ν ( dx ′ ), • σ i ( y, ν ) = R ˜ σ i ( y, x ′ ) ν ( dx ′ ), • d ( x, y, ν, h ) = R ˜ d ( x, y, x ′ , h ) ν ( dx ′ ).Then it is easy to check that for this example Condition 2.1 is satisfied. One can also check thatConditions 6.2, 7.1 and 7.4 are satisfied as well. In particular, note that for x ∈ R d , y, y ′ ∈ R m , imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles ν, ν ′ ∈ P ( R m ), b ( x, y ′ , ν ′ ) − b ( x, y, ν ) = Z (˜ b ( x, y, x ′ )( ν ′ − ν )( dx ′ ) + Z (˜ b ( x, y ′ , x ′ ) − ˜ b ( x, y, x ′ )) ν ( dx ′ )+ Z (˜ b ( x, y ′ , x ′ ) − ˜ b ( x, y, x ′ ))( ν ′ − ν )( dx ′ ) . Using Taylor’s expansion to the second term we get, Z (˜ b ( x, y ′ , x ′ ) − ˜ b ( x, y, x ′ )) ν ( dx ′ ) = ( y ′ − y ) Z ˜ b y ( x, y, x ′ ) ν ( dx ′ ) + 12 ( y ′ − y ) r ( x, y, y ′ ) , where r is a bounded function. Using Taylor’s expansion to the third term we get Z (˜ b ( x, y ′ , x ′ ) − ˜ b ( x, y, x ′ ))( ν ′ − ν )( dx ′ ) = ( y ′ − y ) Z ˜ b y ( x, y, x ′ )( ν ′ − ν )( dx ′ )+ 12 ( y ′ − y ) r ( x, y, y ′ ) , where r is a bounded function. Finally using the boundedness and continuity of ˜ b , ˜ b y and the inequality | ( y ′ − y ) Z ˜ b y ( x, y, x ′ )( ν ′ − ν )( dx ′ ) | ≤ | y ′ − y | + | Z ˜ b y ( x, y, x ′ )( ν ′ − ν )( dx ′ ) | we see that Condition 7.1 is satisfied. Conditions 6.2,7.4 can be verified similarly. Define for t ∈ [0 , T ], the function f t from D R d +2 m ×P ( R d ) [0 , t ] to R d as follows:For ( x (1)[0 ,t ] , x (2)[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] ) = ζ [0 ,t ] ∈ D R d + m ×P ( R d ) [0 , t ] f t ( ζ [0 ,t ] ) = b c ( x (1) t , y t , ν t , x (2) t ) + b ( x (1) t , y t , ν t ) s ,t ( ζ (2)[0 ,t ] ) , (7.35)where as before ζ (2) = ( x (2) , y, w, ν ). We note that k f t ( ζ [0 ,t ] ) k = s ,t ( x (1) t , x (2)[0 ,t ] , ζ (2)[0 ,t ] ) + s ,t ( x (1) t , x (2)[0 ,t ] , ζ (2)[0 ,t ] ) + s ( x (1) t , x (2) t , x (2) t , y [0 ,t ] , ν [0 ,t ] ) . (7.36)Also define for t ∈ [0 , T ], the function ¯ f t from D R d +2 m ×P ( R d ) [0 , t ] × R + × R d to R as follows: For( x (1)[0 ,t ] , x (2)[0 ,t ] , y [0 ,t ] , w [0 ,t ] , ν [0 ,t ] , u, h ) = ( ζ [0 ,t ] , u, h ) ∈ D R d + m ×P ( R d ) [0 , t ] × R + × R d ¯ f t ( ζ [0 ,t ] , u, h ) = 1 [0 ,d ( x (1) t − ,y t − ,ν t − ,h )] ( u ) 1 d ( x (1) t − , y t − , ν t − , h ) (cid:16) d c ( x (1) t − , y t − , h, ν t − , x (2) t − )+ s ,t ( ζ (2)[0 ,t ] ) · d ( x (1) t − , y t − , h, ν t − ) (cid:17) . The functions f t , ¯ f t will play the role of kernels for certain integral operators on L spaces. To describethese operators, in addition to the canonical spaces and processes introduced in Section 2.2 (see (2.8),(2.9)), we define the canonical processes V ∗ = ( B ∗ , n ∗ , Y ∗ ) on Ω m as V ∗ ( ω ) = ( B ∗ ( ω ) , n ∗ ( ω ) , Y ∗ ( ω )) = ( ω , , ω , , ω , ); ω = ( ω , , ω , , ω , ) ∈ Ω m . Also, with Π as introduced in Remark 2.1, let µ ∗ : Ω m → D P ( R d ) [0 , T ] be defined as µ ∗ ( ω ) =Π( Y ∗ ( ω ) , B ∗ ( ω ) , n ∗ ( ω )). Write V ∗ = ( Y ∗ , B ∗ , µ ∗ ) . (7.37) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles We can now define the integral operators related to f t and ¯ f t . Recall the transition probability kernel α introduced in (2.7). Fix ω ∈ Ω m and consider the Hilbert space H ω = L ( Ω d , α ( ω , · )). We denotethe norm and inner product in H ω as k · k ω and h· , ·i ω respectively. Define the integral operator A ω on H ω as follows. For g (1) ∈ H ω , ( A ω g (1) ) = ˆ g (1) ω , where for ω ∈ Ω d ,ˆ g (1) ω ( ω ) = Z Ω d g (1) ( ω ) Z T f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dB ∗ ,t ( ω ) ! α ( ω , dω ) . Also define the integral operator A ω on H ω as follows. For g (2) ∈ H ω , ( A ω g (2) ) = ˆ g (2) ω , where for ω ∈ Ω d ,ˆ g (2) ω ( ω ) = Z Ω d g (2) ( ω ) (cid:18)Z X T ¯ f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ) , u, h ) d ˜ n ∗ ( ω ) (cid:19) α ( ω , dω ) . Let A ω = A ω + A ω . Denote by I the identity operator on H ω . Lemma 7.8. For P a.e. ω , (i) Trace ( A ω ( A ω ) ∗ ) = 0 ; (ii) Trace ( A nω ) = 0 for all n ≥ ; and (iii) I − A ω is invertible. Proof. Parts (i) and (ii) are consequences of independence between B ∗ and n ∗ under α ( ω , · ). Forexample for (i), from the definitions of A iω , it follows thatTrace( A ω ( A ω ) ∗ ) = Z Ω d Z T f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dB ∗ ,s ( ω ) !(cid:18)Z X T ¯ f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ) , u, h ) d ˜ n ∗ ( ω ) (cid:19) α ( ω , dω ) α ( ω , dω ) . The above expression is 0 due to the independence between B ∗ and n ∗ under α ( ω , · ). Part (ii) is provedsimilarly (see e.g. Lemma 2.7 of [20] ). Part (iii) is now immediate from Lemma 1.3 of [20]. J N, and J N, . Recall the integral operators A iω , i = 1 , 2, introduced in Section 7.4. Define τ ( i ) : Ω m → R as τ ( i ) ( ω ) =Trace( A iω ( A iω ) ∗ ), i = 1 , 2. From Lemma 7.8 we have that, for P a.e. ω ,Trace( A ω ( A ω ) ∗ ) = τ (1) ( ω ) + τ (2) ( ω ) . (7.38)The following lemma gives the asymptotics for the second terms on the right sides of (7.6) and (7.23). Lemma 7.9. As N → ∞ , N N X j =1 Z T m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] ) dt − τ (1) ( V ) and N N X j =1 Z [0 ,T ] × R d ¯ m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt − τ (2) ( V ) converge to in probability. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles Proof. Note that if A is an integral operator on L ( ν ) with associated kernel a ( x, y ), then Trace( AA ∗ ) = || a || L ( ν ⊗ ν ) . Thus from the definition of the operator A ω ,Trace( A ω ( A ω ) ∗ )= Z Ω d (cid:12)(cid:12)(cid:12)(cid:12) Z T f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dB ∗ ,t ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) α ( ω , dω ) α ( ω , dω )= Z Ω d Z T Z Ω d k f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) k α ( ω , dω ) dt α ( ω , dω )Using the relation (7.36) we have,Trace( A ω ( A ω ) ∗ ) = Z Ω d Z T Z Ω d (cid:16) s ,t ( X ∗ ,t ( ω ) , X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ))+ s ,t ( X ∗ ,t ( ω ) , X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ))+ s ( X ∗ ,t ( ω ) , X ∗ ,t ( ω ) , X ∗ ,t ( ω ) , V ∗ ,t ( ω )) (cid:17) α ( ω , dω ) dt α ( ω , dω )= Z Ω d Z T m t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dt α ( ω , dω ) . Since conditional on G , { X j } are i.i.d. with common distribution α ( V , · ) ◦ X − ∗ , the first convergencein the lemma now follows from the weak law of large numbers. The second convergence statement isproved similarly.We will now use the results from Section 4 with X = Ω d and ν = α ( ω , · ), ω ∈ Ω m . For each ω ∈ Ω m , k ≥ f ∈ L sym ( α ( ω , · ) ⊗ k ) the multiple stochastic integral I ω k ( f ) is defined as inSection 4. More precisely, let A p be the collection of all measurable f : Ω m × Ω pd → R such that Z Ω p | f ( ω , ω , . . . , ω p ) | α ( ω , dω ) · · · α ( ω , dω p ) < ∞ , P a.e. ω and f ( ω , · ) is symmetric for P a.e. ω . Then there is a measurable space ( Ω ∗ , F ∗ ) and a regularconditional probability distribution α ∗ : Ω × F ∗ → [0 , 1] such that on the probability space ( Ω m × Ω ∗ , B ( Ω m ) ⊗ F ∗ , P ⊗ α ∗ ), where P ⊗ α ∗ ( A × B ) = Z A α ∗ ( ω , B ) P ( dω ) , A × B ∈ B ( Ω m ) ⊗ F ∗ , there is a collection or real valued random variables { I p ( f ) : f ∈ A p , p ≥ } with the properties that(a) For all f ∈ A the conditional distribution of I ( f ) given G ∗ = B ( Ω m ) ⊗ {∅ , Ω ∗ } is Normal withmean 0 and variance R Ω d f ( ω , ω ) α ( ω , dω ).(b) I p is (a.s.) linear map on A p .(c) For f ∈ A p of the form f ( ω , ω , . . . , ω p ) = p Y i =1 h ( ω , ω i ) , s.t. Z Ω d h ( ω , ω ) α ( ω , dω ) < ∞ , P a.e. ω ,I p ( f ) = ⌊ p/ ⌋ X j =0 ( − j C p,j (cid:18)Z Ω d h ( ω , ω ) α ( ω , dω ) (cid:19) j ( I ( h )) p − j imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles and Z Ω ∗ ( I p ( f )( ω , ω ∗ )) α ∗ ( ω , dω ∗ ) = p ! (cid:18)Z Ω d h ( ω , ω ) α ( ω , dω ) (cid:19) p P a.e. ω . We write I p ( f )( ω , · ) as I ω p ( f ). With an abuse of notation, we will denote once more by V ∗ the canonical process on Ω m × Ω ∗ , i.e. V ∗ ( ω , ω ∗ ) = ω , for ( ω , ω ∗ ) ∈ Ω m × Ω ∗ .Recall the class A introduced in Section 2.2. Let for ϕ ∈ AV ϕN = √ N N N X j =1 ϕ ( X j ) − m ϕ ( V ) . Define ¯ τ : Ω m × Ω ∗ → R as ¯ τ ( ω , ω ∗ ) = Trace( A ω ( A ω ) ∗ ).Given ω ∈ Ω m , define F ω : Ω d × Ω d → R as follows: For ( ω , ω ) ∈ Ω d × Ω d F ω ( ω , ω ) = Z T f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dB ∗ ,t ( ω )+ Z T f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dB ∗ ,t ( ω ) − Z T m t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) dt. Also, given ω ∈ Ω m , define F ω : Ω d × Ω d → R as follows: For ( ω , ω ) ∈ Ω d × Ω d F ω ( ω , ω ) = Z X T ¯ f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ) , u, h ) d ˜ n ∗ ( ω )+ Z X T ¯ f t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω )) d ˜ n ∗ ( ω ) − Z [0 ,T ] × R d ¯ m t ( X ∗ , [0 ,t ] ( ω ) , X ∗ , [0 ,t ] ( ω ) , V ∗ , [0 ,t ] ( ω ) , h ) γ ( dh ) dt, where ˜ n ∗ is the compensated PRM: ˜ n ∗ = n ∗ − ν .Also let F : Ω m × Ω d × Ω d → R be defined as F ( ω , ω , ω ) = 12 (cid:0) F ω ( ω , ω ) + F ω ( ω , ω ) (cid:1) , ( ω , ω , ω ) ∈ Ω m × Ω d × Ω d . Let σ N ( F ) = N X i,j =1 i = j F ( V , V i , V j ) . From Lemmas 7.2, 7.3, 7.5, 7.6 and 7.7 it follows that J N, ( T ) + J N, ( T ) = N − σ N ( F ) − N N X j =1 Z T m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] ) dt − N N X j =1 Z [0 ,T ] × R d ¯ m t ( X j [0 ,t ] , X j [0 ,t ] , V [0 ,t ] , h ) γ ( dh ) dt (cid:17) + R N , (7.39)where R N converges to 0 in probability. imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles In order to study the asymptotics of the expression on the left side of (5.2), we need to consider thejoint asymptotic behavior of V ϕN and N − σ N ( F ). Denote by ℓ Nϕ the measurable map from Ω m to P ( R )such that L (cid:0) ( V ϕN , N − σ N ( F )) | G (cid:1) = ℓ Nϕ ( V ) , a.s.Next note that F ∈ A and so I ( F ) is a well defined random variable on ( Ω m × Ω ∗ , B ( Ω m ) ⊗F ∗ , P ⊗ α ∗ ). Also define ¯Φ : Ω m × Ω d → R as¯Φ( ω , ω ) = Φ ω ( ω ) = ϕ ( X ∗ ( ω )) − m ϕ ( ω ) . Note that ¯Φ ∈ A and so I ( ¯Φ) is well defined. Let ℓ ϕ be a measurable map from Ω m to P ( R ) suchthat L (cid:0) ( I ( ¯Φ) , I ( F )) | G ∗ (cid:1) = ℓ ϕ ( V ∗ ) . From Theorem 4.1 it follows that ℓ Nϕ ( ω ) → ℓ ϕ ( ω ) weakly for P a.e. ω . (7.40)The following lemma is the key step. Lemma 7.10. As N → ∞ , i V ϕN + J N, ( T ) + J N, ( T ) converges in distribution to iI ( ¯Φ) + I ( F ) − ¯ τ . Proof. From (7.39) and Lemma 7.9 we have that i V ϕN + J N, ( T ) + J N, ( T ) = i V ϕN + N − σ N ( F ) − 12 ( τ (1) ( V ) + τ (2) ( V )) + ˜ R N , where ˜ R N converges to 0 in probability. There are measurable maps ζ N , ζ from Ω m to P ( C ), where C is the complex plane, such that with S N = i V ϕN + N − σ N ( F ) − 12 ( τ (1) ( V ) + τ (2) ( V ))and S = iI ( ¯Φ) + I ( F ) − 12 ¯ τ L ( S N | G ) = ζ N ( V ) , L ( S | G ∗ ) = ζ ( V ∗ ) . From (7.40) and the definitions of τ ( i ) and ¯ τ , ζ N ( ω ) → ζ ( ω ) , weakly for P a.e. ω . (7.41)Finally, denote the probability distribution of ( V , S N ) on Ω m × C by ρ N and that of ( V ∗ , S ) on Ω m × C by ρ . Then ρ N and ρ can be disintegrated as ρ N ( A × B ) = Z A ζ N ( ω )( B ) P ( dω ) , ρ ( A × B ) = Z A ζ ( ω )( B ) P ( dω ) , for A ∈ B ( Ω m ), B ∈ B ( C ). From (7.41) it now follows that ρ N → ρ weakly. The result follows. Recall the operator A ω introduced in Section 7.4 and let Φ ω be as in (2.10). Define for ω ∈ Ω m , σ ϕω = k ( I − A ω ) − Φ ω k L (Ω d ,α ( ω , · )) . (7.42) imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles It follows from Lemma 1.2 of [20] and Lemma 7.8 that P a.s. E [exp( 12 I ( F )) | G ∗ ] = exp( 12 Trace( A V ∗ ( A V ∗ ) ∗ ))where E is the expectation operator on ( Ω m × Ω ∗ , B ( Ω m ) ⊗ F ∗ , P ⊗ α ∗ ). Therefore E exp (cid:18) I ( F ) − 12 ¯ τ (cid:19) = 1 . Also, recall that E P N exp (cid:0) J N, ( T ) + J N, ( T ) (cid:1) = 1 . Now applying Lemma 7.10 with ϕ ≡ J N, ( T ) + J N, ( T )) is uniformly integrable. Also since | exp( i V ϕN ) | = 1,exp( i V ϕN + J N, ( T ) + J N, ( T ))is uniformly integrable as well. Using Lemma 7.10 again we have thatlim N →∞ E P N (cid:2) exp( i V ϕN + J N, ( T ) + J N, ( T )) (cid:3) = E (cid:20) exp( iI ( ¯Φ) + 12 I ( F ) − 12 ¯ τ ) (cid:21) = E (cid:20) E (cid:18) exp( iI ( ¯Φ) + 12 I ( F ) − 12 ¯ τ ) | G ∗ (cid:19)(cid:21) = Z Ω m exp (cid:18) − 12 ( σ ϕω ) (cid:19) P ( dω ) , where the last equality is a consequence of Lemma 1.3 of [20] and Lemma 7.8. Thus we have proved(5.2) which completes the proof of Theorem 2.4. 8. Convergence of the Signed Measures in the Path Space. In [14] authors studied a functional central limit theorem for scaled and centered empirical measuresfor a family of weakly interacting particle systems with a common factor. As noted in the Introduction,in the current work our focus is on limit theorems for functionals of the whole path of the particles,however in this section we will discuss how functional central limit theorems of the form in [14] can berecovered from Theorem 2.4. For t ∈ [0 , T ] consider the random signed measure on R d defined asΛ Nt = √ N N N X j =1 δ Z N,jt − µ t . (8.1)We note that µ t = η t ( ¯ V ) where ¯ V is as introduced in Section 2.2 and for ω ∈ Ω m , η t ( ω ) = α ( ω , · ) ◦ X − ∗ ,t with α as in (2.7) and X ∗ as in (2.9).For notational simplicity we assume for rest of the section that d = 1. Following [9] and [14] Λ N = { Λ Nt } t ∈ [0 ,T ] can be regarded as a sequence of D Ψ ′ [0 , T ] valued random variables where Ψ ′ is the dual ofthe “modified Schwartz space” Ψ given as follows. Let ρ : R → R be defined as ρ ( x ) = C exp {− / (1 − | x | ) } | x | < , x ∈ R , imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where C ∈ (0 , ∞ ) is such that R ρ ( x ) dx = 1. Let v ( x ) = Z e −| y | ρ ( x − y ) dy, e ( x ) = 1 /v ( x ) , x ∈ R . Let Ψ = { ψ = eu : u ∈ S} where S is the Schwartz space (cf. [7]). For p ∈ N and ψ ∈ Ψ, define k ψ k p = X ≤ k ≤ p Z R (1 + | x | ) k (cid:12)(cid:12)(cid:12)(cid:12) d k dx k ( ψ ( x ) v ( x )) (cid:12)(cid:12)(cid:12)(cid:12) dx. Let Ψ p be the completion of Ψ with respect to k · k p . The Ψ p is a Hilbert space with inner product h· , ·i p defined in an obvious manner. For ˆ φ ∈ Ψ and φ ∈ Ψ p ˆ φ [ φ ] . = Z R ˆ φ ( x ) φ ( x ) v ( x ) dx defines a continuous linear functional on Ψ p with norm k ˆ φ k − p = sup φ ∈ Ψ p | ˆ φ [ φ ] |k φ k p . Let Ψ − p be the completion of Ψ with respect to this norm. Then Ψ is a nuclear space [7] and Ψ ′ . = ∪ ∞ k =0 Ψ − k is its dual.Recall the class A introduced in Section 2.2. Given ℓ ∈ N , t , · · · t ℓ ∈ [0 , T ] and φ , · · · φ ℓ ∈ Ψ, define ϕ i ∈ A , i = 1 , · · · , ℓ as ϕ i ( ω ) = φ i ( ω t i ), ω ∈ D d . Also, for ω ∈ Ω m , let Φ iω = ϕ i ( X ∗ ) − m ϕ i ( ω ) where m · is as introduced in Section 2.2. Also define the ℓ × ℓ matrix Σ ω = (Σ ijω ), whereΣ ijω = h ( I − A ω ) − Φ iω , ( I − A ω ) − Φ jω i L (Ω d ,α ( ω , · )) . Let γ ϕ ω be a ℓ dimensional Gaussian random variable with mean 0 and variance Σ ω and define γ φ , ··· ,φ ℓ t , ··· t ℓ ≡ γ ϕ = Z Ω m γ ϕ ω P ( dω ) . The following theorem follows from Theorem 2.4 of the current work and arguments similar to Theorem3.1 of [14]. We only provide a sketch. Let Q N ∈ P ( D ([0 , T ] : Ψ ′ )) be the probability law of Λ N . DefineΠ φ , ··· ,φ ℓ t , ··· t ℓ : D Ψ ′ [0 , T ] → R ℓ as Π φ , ··· ,φ ℓ t , ··· t ℓ ( u ) = ( u t [ φ ] , · · · , u t ℓ [ φ ℓ ]) . Theorem 8.1. Suppose all the assumptions in Theorem 2.4 are satisfied. Then, as N → ∞ , Q N → Q where Q is the unique probability measure on D Ψ ′ [0 , T ] that satisfies Q ◦ (Π φ , ··· ,φ ℓ t , ··· t ℓ ) − = γ φ , ··· ,φ ℓ t , ··· t ℓ for all ℓ ≥ , t , · · · , t ℓ ∈ [0 , T ] and φ , · · · φ ℓ ∈ Ψ . Sketch of Proof. From Theorem 4.1 and Proposition 5.2 of [18] it suffices to show that(i) for every φ ∈ Ψ, Q Nφ is tight in D R [0 , T ], where Q Nφ = Q N ◦ (Π φ ) − and Π φ : D Ψ ′ [0 , T ] → D R [0 , T ] isdefined as Π φ ( u )[ t ] = u t [ φ ], t ∈ [0 , T ].(ii) for all ℓ ≥ t , · · · , t ℓ ∈ [0 , T ] and φ , · · · φ ℓ ∈ Ψ, Q N ◦ (Π φ , ··· ,φ ℓ t , ··· t ℓ ) − → γ φ , ··· ,φ ℓ t , ··· t ℓ .Proof of (i) follows along the lines of Theorem 3.1 of [14] and is omitted. Consider now (ii). Fix ℓ ≥ t , · · · , t ℓ ∈ [0 , T ] and φ , · · · φ ℓ ∈ Ψ as above. Let a , · · · a l ∈ R and define ϕ a = ℓ X i =1 a i ϕ i , Φ a ω = ℓ X i =1 a i Φ iω , imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles where ϕ i , Φ iω are as defined above the theorem. Let σ a ω = k ( I − A ω ) − Φ a ω k L (Ω d ,α ( ω , · )) and ˜ γ a ω be a Normal random variable with mean 0 and variance ( σ a ω ) and let ˜ γ a = R ˜ γ a ω P ( dω ). Let ˜Π a : D Ψ ′ [0 , T ] → R be defined as ˜Π a ( u ) = P ℓi =1 a i u t i [ φ i ]. From Theorem 2.4 it is immediate that Q N ◦ ( ˜Π a ) − → ˜ γ a as N → ∞ . The statement in (ii) is now immediate from the classical Cram´er-Woldargument. 9. Application to Finance Recently in [2], authors have introduced a model for self-exciting correlated defaults in which defaulttimes of various entities depend not only on factors specific to entities and a common factor but alsoon the average number of past defaults in the market. The paper studies an asymptotic regime as thenumber of entities become large. One of the results in [2] is a CLT which is established under somewhatrestrictive conditions on the model. Below, we describe the result from [2] and then remark on how theresults of current paper provide a CLT for the model in [2] under much lesser restrictive conditions andfor some of its variations.The model for which CLT is considered in [2] (see Section 5.3 therein), using notation of the currentpaper, is as follows. Let ( B i ) i ∈ N be a sequence of real standard Brownian motions and let ( n i ) i ∈ N be asequence of Poisson random measures on X T = [0 , T ] × R × R + with intensity measure ν = λ T ⊗ δ { } ⊗ λ ∞ ,given on a filtered probability space ( Ω, F , P , {F t } ). All these processes are mutually independent andthey have independent increments with respect to the filtration {F t } . Consider the system of equationsgiven by U Nt = U + R t β ( U Ns , ¯ L Ns ) ds + R t ¯ σ ( U Ns , ¯ L Ns ) dB s ,X N,it = X N,i + R t β ( X N,is , Y N,is , U Ns , ¯ L Ns ) ds + B it , i = 1 , . . . , N ,Y N,it = R X t [0 ,λ ( X N,is ,Y N,is − ,U Ns , ¯ L Ns − )] ( u ) n i ( ds dh du ) , (9.1)where ¯ L Nt = N N X i =1 ζ ( Y N,it ) for some bounded and Lipschitz map ζ , and we assume that { X N,i } Ni =1 arei.i.d. with common distribution µ and U is independent of { X N,i } Ni =1 and has probability distribution ρ . Also, { X N,i } Ni =1 and U are F measurable. The interpretation for the finance model is as follows.There are N defaultable firms. The process U N represents the common factor process and X N,i isthe i -th firm’s specific factor. Y N,i are counting processes representing the number of defaults of firm i . The key feature of this model is that the correlation among the defaults not only depends on thecommon exogenous factor U N , but also on the past defaults through the process ¯ L N . In the model of[2], ζ ( y ) = | y | ∧ Y N,it greater than 0 are treated the same way (an entityhas either not defaulted by time t or it has defaulted in which case it disappears from the system.) Thepaper [2] establishes a CLT for ¯ L Nt under the condition that λ ( x, y, u, l ) ≡ λ ( l ), x, y, u, l ∈ R . Note thatin this case the factor processes X N,i and U N become irrelevant.The model in (9.1) is a special case of the model considered in (2.1) and (2.2) with the followingidentifications: • d = 2, m = 1. • Z N,i = ( X N,i , Y N,i ) ′ . • b = ( ¯ β, ′ , b = ¯ β , where for z ∈ R , u ∈ R , ν ∈ P ( R ),¯ β ( z, u, ν ) = β ( z, u, h ˆ ζ, ν i ) , ¯ β ( u, ν ) = β ( u, h ˆ ζ, ν i ) , where ˆ ζ : R → R is defined as ˆ ζ ( x, y ) = ζ ( y ), ( x, y ) ∈ R . imsart-generic ver. 2011/11/15 file: mainrev2.tex date: September 16, 2018 CLT for Weakly Interacting Particles • σ ( u, ν ) = ¯ σ ( u, h ˆ ζ, ν i ), σ = (cid:18) (cid:19) (See Remark 2.2). • d = 0, d = ¯ λ , where for z ∈ R , u ∈ R , ν ∈ P ( R ), ¯ λ ( z, u, ν ) = λ ( z, u, h ˆ ζ, ν i ).Coefficients β, β and λ are required to satisfy the following conditions. (A1) The function β is bounded and Lipschitz. β ( z, u, l ) is twice continuously differentiable in u and l with bounded derivatives. (A2) The function β is bounded and Lipschitz. β ( u, l ) is twice continuously differentiable in u and l with bounded derivatives. Exactly same assumptions for ¯ σ (A3) The function λ is nonnegative, bounded, Lipschitz and it is bounded away from 0. λ ( z, u, l ) istwice continuously differentiable in u and l with bounded derivatives.Under the above assumptions it can be easily checked that Conditions 2.1, 6.2, 7.1, 7.4 and themodified form of Condition 2.3 in Remark 2.2 are satisfied. Thus from Theorem 2.4 it follows that theaverage default process { ¯ L Nt } satisfies a CLT. More precisely, for t ∈ [0 , T ], √ N ( ¯ L Nt − m t ( B , U ))converges in distribution to a random variable whose distribution is given as a mixture of Gaussians,where for t ∈ [0 , T ], m t : C × R → [0 , 1] is the measurable map such that m t ( B , U ) = E ( ζ ( Y t ) | B , U )if ( U, X, Y, α ) solve the following nonlinear system of equations. 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