increasing propagation of chaos for mean field modelsbenarous/publications/benarous_35.pdf · 2007....

Ann. Inst. Henri PoincarP,

Vol. 35, no I, 1999, p. 85-102 Probabilite’s et Statistiques

Increasing propagation of chaos for mean field models

G. BEN AROUS * DMI, Ecole Normale Superieure, 45, rue d’Ulm, 75230 Paris Cedex 05, France

and

0. ZEITOUNI + Department of Electrical Engineering, Technion- Israel Institute of Technology, Haifa 32000, Israel

ABSTRACT. - Let /&N) denote a mean-field measure with potential F. Asymptotic independence properties of the measure p(N) are investigated. In particular, with H(.I,u) denoting relative entropy, if there exists a unique non-degenerate minimum of H(. 1~) - F( .), then propagation of chaos holds for blocks of size o(N). Certain degenerate situations are also studied. The results are applied for the Langevin dynamics of a system of interacting particles leading to a McKean-Vlasov limit. 0 Elsevier, Paris

Key words: Large deviations, Gibbs potential, exchangeability.

RWJMG. - Soit pcN) une mesure de type champ-moyen avec potentiel d’interaction F. Les proprietes asymptotiques d’independance de la mesure p(N) sont Ctudiees. En particulier, si H(.Ip) designe l’entropie relative, on montre que, s’il existe un unique minimum non dCgCnCre de H(.Ip) - F(.), alors la propagation du chaos est valide pour les blocs de taille o(N). Certains cas de minima dCgCnCr& sont aussi Ctudies. Les resultats

AMS Subject classi&zrion: Primary 60FlO. Secondary 60G09, 6OK35. * Current address: Dept. of Mathematics, Ecole Polytechnique Fed&ale, CH-1015 Lausanne,

Suisse. .’ The work of this author was partially supported by a US-Israel BSF grant.

Anna/es de I’lnsti~ Hem-i Poincur6 ProbabilitCs et Statistiques 0246.0203

Vol. 35/99/01/O Elsevier, Paris

86 G. BEN AROUS AND 0. ZEITOUNI

sont appliques a la dynamique de Langevin d’un systeme de particules convergeant vets une limite de McKean-Vlasov. 0 Elsevier, Paris

1. INTRODUCTION

Let (S, S, p) denote a (Polish) measure space, let F : Ml(S) -+ IR be measurable and bounded, and let X = X1 j , X,V denote a sequence of random variables distributed according to the mean jield Gibbs measure with potential F

,dx)(dzl,. . , dxR’) = Z,’ exp(NF(L”,)) b p(dx) . (1) i=l

Here Lk = $ CFl, S,r, is the empirical measure of the vector x = {:ci}~YI, and ZN = JsAv ~x~(NF(L;~)) n,“l, p(dz,) is a normalization constant. In various places, we also use L$ := N-l C,“l, 6-x-, to denote the (random) empirical measure of the random sample X.

Under mild assumptions, the law of L$ converges in distribution, under the law ~(~1, to a deterministic measure ~5,~. By exchangeability, this implies the convergence of the law of X1 under IJ,(~) to p*, and more generally, for any k finite, the convergence of the law of (XI, . . . ! XL.) under ~(~1 to (/I*)@‘.

For any exchangeable measure 19(“) on S”, let Oiv,’ denote its marginal on the first k coordinates; that is, for A c S” measurable,

By exchangeability, for any permutation (T : { 1 i . . , N} + { 1, . . . N},

i.e. the k-marginals of /A civ) do not depend on the choice of coordinates. Recall that the relative entropy H(.l.) is defined as

if v < LL otherwise.

INCREASING PROPAGATION OF CHAOS 87

In this article, we estimate the relative entropy distance between p(N) and appropriate (simpler) exchangeable measures v(~), which are related to the law h*. The main interest in obtaining such estimates stems from the fact that if one has that

BN = H(p(N)~v(~))

then, for 1(N) = yBN a product measure, and IF(N) = o(N/Bn;),

(See e.g. [5, (2.10)] for the first inequality). Hence, for appropriate (large, increasing) blocks k(N), the relative entropy distance (and hence, also the variational distance) between hN,k(N) and yXlccN) converges to 0. This implies a strong version of the propagation of chaos. It is important to notice that the notion of convergence we use is well suited to deal with increasing blocks: a statement in the weak topology of Ml(S”) would not be an advance over the finite dimensional propagation of chaos.

We remark that one can, by consideration of the function F(V) = g(/ zdv) with smooth g( .), provided the latter integral is well defined and that the support of I-L is bounded, adapt the set-up described above to Gibbs measures involving empirical means, as opposed to empirical measures. See the remark at the end of Section 2.

Similarly, the ideas presented here can be adapted to the Gaussian setup, where they can be used to yield sharper versions of CLT convergence. For a discussion of this application, we refer the reader to [2].

As will be seen, the critical value of k(N) and the structure of v(~“) depend crucially on the behaviour of the function H(.~/L) - F(.) near its minima. In particular, if the minimum is unique, say /L*, then I/(~) = (IL*)%~, with k(N) = o(N) if this minimum is non- degenerate. Thus, with a non-degenerate unique minimum, one concludes the propagation of chaos for blocks of size o(N).

This study is related to the one in [6], where a similar question in the case of Gibbs conditioning was considered. Due to the extra regularity provided by integration over F, the results here are more satisfactory in that they cover (with sharp rates) genuinely infinite dimensional situations.

The structure of this article is as follows. In the rest of this introduction, we describe our results for a (simple) problem, the Curie-Weiss model. This model exhibits already a range of interesting phenomena. We describe the precise assumptions we put on the function F. the statement of our

Vol. 35. n” l-1999

88 G.BEN AROUS AND 0. ZEITOUNI

main results in the non-degenerate case, and an application to the Langevin dynamics of interacting particles, in Section 2. Section 3 is devoted to the proofs of the non-degenerate case, while Section 4 is devoted to the statements and proofs for the degenerate case. Turning now to the Curie-Weiss model, let S = { -1: l}, /I = i(n-1 + &), F(V) = F,,,ij(~) = /Y(v.x:)’ + ~(Y.:x) and

We distinguish between the following cases

I. 0 = 0, /j < Jj or 0 # 0 II. N = 0, p = +

III. o! = 0, /j > i

Let I(Y) = H(v\/J) - FQ,;s(~/). A degenerate minima in this context is a minimum IL* with 1”(p*) = 0. It is easy to check (by embedding Ml(S) into R) that in both cases I and II, IL* is unique, with case I corresponding to a non-degenerate LL* while case II is degenerate. On the other hand, case III corresponds to the case of two non-degenerate minima p;, ,Q; (in all cases, and for any minimum p*, p* zp* (1) satisfies the relation log(,~*/( 1 - ,i~*)) = 4p(2fi* - 1) + 2a,!i*, with i/* = l/2 for <Y = 0 and p 5 l/2). A corollary of our general results in this paper is the

COROLLARY 1.

Case I. - Let k(N) = o(N). Then

Case II. - Let k(N) = o(Nt). Then

Case III. - Let k(N) = o(N). Then

The possibility of working with k = o(N) in case III was pointed out to us by A. Dembo, who also provided a proof based on the general results [7].


For a study, from a different viewpoint, of the statistics of large blocks of variables in the critical case, we refer the reader to [lo] and the references therein. F. Comets has kindly pointed out to us that the rate k(N) = o(Ni ) in the critical case II is optimal. Indeed, fix 1 > t > 0, and let It(N) = tv’% and 0 N := IC(N)-l/’ EFLy’ X;. Then, by [lo, Theorem l] (taking there p = fi and Q = me-l), there exists a random variable WN such that the law, under ,u~.~(~), of the random vector ( WN, ON - &N) converges weakly to a non-degenerate product measure, with first marginal non- Gaussian and second marginal Gaussian. In particular, @N possesses a non-Gaussian limit law for any t > 0. On the other hand, under the law @“(“) the standard CLT implies that its limit law is Gaussian.

Although Corollary 1 is a direct application of our general Theorems 1-4, we provide a (direct) proof of parts I and II in the end of Section 4. Thus, the reader interested in understanding first this simple situation may skip directly there, avoiding the use of the results in [3] or 141.

ACKNOWLEDGMENT

We thank A. Dembo for useful suggestions concerning the multiple minima situation, and an anonymous referee for useful comments which led to a simplification of our arguments.

2. THE NON-DEGENERATE CASE

To state our results, we need to introduce some notations, following [ 11. Let J(Y) = H(v/,u) - F(v). Denote m, = infVEnf,(s) I(V). Let

ICF = {v E M,(S) : H(vlp) - F(v) = m,}.

Because F is bounded and continuous, KCF is nonempty and compact. We often need the assumption of uniqueness of minimizers, summarized as: (Al) There exists a unique ,LL* E ICF. (See however Theorem 2 for a discussion of the case where (Al) does not hold).

Let Ek(S) = &kCb(S), the k-fold (projective) tensor product of C&(S). Let E,(S) = @g”=,&(S), and let sE,( S) denote the symmetric sub- algebra of E,(S) . For any e E E,(S), deg( e) is the largest integer ,% 2 1 with nonzero component in the sum defining e.

Vol. 3s. Ilo l-1999


Let V E s&(S), with &g(V) = 7’. 7’ 2 2, and components If,: X: = 2,... r. Let

F(I/) = -g(Vk, IP) . I,.=2

(We do not need to consider linear components in the definition of F(.). for these can be eliminated by a suitable modification of the measure /L).

Note that

NF(LLiv) = c fl-(“-l) c Vk(X,, . . . 7’. ) . “‘0 A.=2 151, <.../~<A7 Define next, on Li(S, p*) (the space of centered, p* square integrable functions), an operator of kernel E such that

Let K(p*) = Ker(IdLgCS,,L ) - E). If K(/L*) = {0}, we say that p* is non-degenerate. The non-degeneracy condition can be also given a Banach space interpretation, c.f. [ 1, Lemma 2.19, pg. 961 and (11) below. Note that by [I, Theorem B], there exist at most a finite number of non-degenerate elements in XF.

Our main resuit is the following.

THEOREM 1. - Assume (Al) and that p* is non-degenerate. Then, for some constant C independent of N,

lim sup H (IriAr) / (p*)%“) < C . (4) ‘V-X

As a consequence, for any k(N) = o(N),

Interesting phenomena occur in the case where ICF consists of a finite number of non-degenerate minima { &, . , PU;}. Let


Define next pUNa* = zf,-, c~(P,‘)@“, that is ,uNl* consists of a mixture of product measures with weights ci. Let pk,* denote the restriction of pN,* to its first Ic coordinates, and note that it is also a mixture of product measures. We now have:

THEOREM 2. - Assume ICF consists of a finite number of non-degenerate minima {&, . . . , p;}. Then, for some constant C which is independent of N,

l$ sop H (piNI 1 pN+) < C . (5) -+

Further, for any k = o(N),

Remark 1. - The constants C appearing in Theorem 1 and Theorem 2 can be computed explicitly, and the statements can be strengthened to yield a convergence to C of the relative entropies in (4) and (5).

2. The results presented above extend immediately if F(LN) is replaced by g(F(LN)), where g(.) E C,“(IR;lB) and F satisfies the assumptions described in the beginning of this section. In this case, I(Y) = HO44 -dFW Th’ is extension allows one to consider interactions based on the empirical mean which are not necessarily polynomial. Technical improvements, in particular on the boundedness assumption on the support of p, are possible but will not be discussed here.

3. By [ 1, Corollary 1.61, one may state the assumptions on the potential F directly in terms of the Banach spaces B appearing in the course of the proof of Theorem 1. We chose not to do so as the introduction of the functions V allows for a rather explicit expression for the non degeneracy condition.

4. As mentioned in the introduction, (6) strengthens the weak convergence announced in [ 1, Theorem B], which by itself precludes the existence of propagation of chaos. Unfortunately, the simple reduction from (5) to (6) used in the case of single minimum does not work in the case of non- product measures, and a slightly more involved argument is needed. The proof given below of (6) is based on a suggestion of A. Dembo.

Finally, we show how to apply Theorem 1 in a dynamic setting to deduce propagation of chaos for a system of interacting particles obeying a Langevin dynamic. Let X$N satisfy the system of SDE’s ]l]

(ix;.” = dB;‘N

Vol. 35, no l-1999.

92 G.BEN AROUS AND O.ZEITOUNI

Here, (Bi>N)l<i<~ denote N independent, R”-valued Brownian motions, U : Rd -+ kRd is a Ct function, and V E I@C~(IR~, IR), which is assumed to admit the representation

v = .I’

c g(.: Tyo2w(dr) .

with (C, w) a compact space, and g : lRd x C + R. differentiable in its first coordinate for all 7 E C. (Note that by [ 1, Corollary 1.61, such a representation with continuous g always exists, and the only additional restriction here involves its differentiability. It holds, e.g., if V(:x, y) possesses a Fourier transform in 1;1(lR2d). Finally, we assume

PO (“)=Law({x;‘“}l<i<,t -1

with 3 E @Cb(lRd, IR), T 2 2. Denote by ,C”,N E AJI(C([O, 7’1; (lRd)“)) the law of (X?“, . . . , X”>“). Next, introduce the McKean-Vlasov nonlinear diffusion: let Bt denote an Rd Brownian motion, independent of x0, and for p E MI(lRd), let

dXt = dz, - VU(X,)dt + .I

VlV(Xt, y)&(dv)dt

Ut = Law(X,). uo=p.

Under the above assumptions, the process x, exists and is well defined [8]. Let FT(p) E AII(C([O,T],IRd)) denote its law. We are now ready to state our main result concerning the dynamics.

THEOREM 3. -Let F(v) = (3, v@“‘), and I(v) = H(vlpo) -F(v). Assume that (Al) holds with p* non-degenerate. Then, there exists a constant C such that

H(CNqF;N(p*)) 2 c.

Further, for k(N) = o(N), propagation of chaos holds, that is

The following corollary is an immediate consequence of Theorem 3.

COROLLARY 2. - Under the assumptions of Theorem 3, let 3 = 0, that is at t = 0 the Xi>N are i.i.d.. Then, with k(N) = o(N),

In particular, at time T independence is still preserved for sub-blocks of size k(N) = o(N).

Anna/es de I’hstifur Henri Poincar~ Probabilitt% et Statistiques


3. PROOFS

Proof of Theorem 1. - Our proof relies on the sharp Laplace asymptotics derived in [l], which in turn build on [3]. By definition,

BN = H(#u(N)I(p*)@N)

S JYL”N)e NF(L”,)&@N =N

L”N, log ~),NF(L~)d~@N

ZN

1% ZN -___ N 1

log ZN rno+N > + J(F(LX,) - F(p*))eNF(L%)dp@N

ZN 1 bB$y + Bj$, (8)

where p’(v) = F(v) - (v,log$).

The sharp asymptotics for the partition function of a mean field model were computed in [ 11. Indeed, [l, Theorem B] yields that

IB$)I=Nmo+ yqNz~log&i&q <cm. (9)

Hence, the proof of Theorem 1 follows as soon as we show that Bg) is uniformly bounded.

By our assumptions, c.f. [l, Theorem 2.13 and Remark 2.161, log $ E C&(S). Hence, F(v) = CI-,=,(Vk,v@“), with VI E Cb(S), and V = (Jfl,.,. , VT) E SE, with deg(V) = T.

We next follow the procedure suggested in [l] in order to embed the computation of Bg) into a Banach space, for which the results of [3] may be applied. By [l], Corollary 1.6, there exists a compact measure space (C, u), a continuous function h : C --+ Cb(S) such that

with V =

I h@‘(+(dT) .

C

Clearly, one also has

Vol. 35, no 1-1999.


for an appropriate v (but same (C,v)). Let B = L”(C. u). B is a type 2 Banach space (c.f. [ 11). Define Th : hfl(S) --f B by

and P : B -+ JR by

With these definitions, F = P o Th. By the fact that pL* is a minimizer (see [ 1, Theorem 2.13 and

Remark 2.16]), it holds that for any neighborhood 0 of p*, there exists a constant K = K(0) such that, whenever L$, E 0,

I@%%) - F(P*))/ L KIWfi(LX, - P*))& . (10)

(See (16) below for an understanding of why (10) holds true). Next, let a denote the law induced by p on B by the map z H h(.)(g). Note that Th,(fi(L$ - p*)) = fi(Th(L$ - h*)), whereas the linearity of Th as a map on Ml(S) implies that Th(Lg - CL*) is an empirical mean of i.i.d.(a), B-valued random variables for which the results of Bolthausen [3] apply. In particular, let (i, H, B) denote the abstract Wiener space generated on B, and denote the Hilbert-norm by H (cf. [3] for details of the construction). Then (cf. [ 1, Lemma 2.19, pg. 96]), the hypothesis of non degeneracy is nothing but the statement that, for $ # 0,

where cp E B*, the dual space of the Banach space B, and $ = Jmp(x)p*(dz) E B.

Returning to the evaluation of BE), note that by (9), for some constant C independent of N, and all N large enough,

B(‘) < C IjJT(j?(L”,) - F(lr*))IeN(~(LXN)-~(p~))d(/d*)~N. N-

.I’

(12)

By Varadhan’s lemma, for any neighborhood 0 of CL*,

,Jiim .I

pv(F(L”,) - F(p*))le N(F(L~,)--F(p.))l(~~,g’O~d(ld*):.N = 0.

(13) Awmler, dr I’hsfitut Hem-i PoinrorP ProbabilitCs et Statistiques


(13) and the fact that the set

{v : IT/JJN(v - p*))IB < JN/c} = {u : IzL(~ - P*)IB < l/cl

is an open set imply that for any constant c > 0,

lir/izp J

IN(F(L”,) - F(/1*))le”(~(LXN)-F(I,*))~(~*)~N

5 l;m!Ep J

]N(qL”,) - ;I(~*))leN(F(LX,)-F(/1*))

ll,~(~(Lx,-r,*))lB<~/c} 4P*FN. (14)

Further, by the bound (lo), for every neighborhood 0 of ,u* and constant c there exists a constant cl = cl(c) such that

(In fact, the precise limit in (15) can be computed by using the CLT of [l], Theorem B, but we do not need it here).

To conclude the proof of Theorem 1, we borrow an argument from [3], pg. 3 15. First, note that by the argument there, there exists an E small enough such that, denoting A, = {z E B : iD2P(Thp*)(z, z)++12 2 l}, one has S(E) := i infzEA, jz]$ > 1. Next, denoting YN = TfL(fi(LX, - p*)) E B, the Frechet differentiability of P implies that

N(F(L”,) -I+*)) = ~(~(Tl(Jq$)) - wdP*N

= ;D2p(Th,L’)(yN; YN) + K(YN) (16)

with K(y)=O.

,~~:o lY12

Let c be such that, for ]?/N]B < a/c,

IN(F(L”,) - F(p*))l 5 ~D2p(Thp*)(yN, YN) + c(yN(i .

Vol. 35, no l-1999.


Then, reducing E further and increasing c, if necessary, one has by Bolthausen’s extension of Yurinskii’s inequality to Banach spaces, c.f. [3, Theorem 31, that, for all c < t < n/c,

(P*)@~(YN E 6%) I exp(-a(c)t).

Hence, for some constant cs independent of N, and all N large enough,

.I {C<lYNl<~l~)

IN(F(L;;.) - p(~*))l,N(r(~~)-~ir‘))d(~*)~N

Se+ tet(p*)‘N(yAv E hA,)dt 5 e + J’

Oc, tetee6(‘)“dt < c2 (17) 0

where the last inequality is due to S(E) > 1. Combining (14), (15) and (17), one concludes that supN BE) < 00,

yielding (4). The second part now follows from the estimate (3). cl

ProofofTheorem 2. - Throughout this proof, C denotes a constant whose value may change from line to line but which is always independent of N. Let

BN = H(pcN)(pN’*).

By definition,

BN=N -?b$ ( i=l

f NEcN) (2 4%4%4 - Jw)) - ; log ($) + t.(LE)) i=l

where throughout E cN) denotes expectations with respect to the measure p(N).

Exactly as in the proof of Theorem 1, one has that B$) is uniformly bounded. Let now 0; denote arbitrary open, disjoint neighborhoods of &, then by [l Theorem B] it holds that P(~)(O;) -‘iv-m ci.

Localizing on Ui=,O; by the large deviations for L$, one finds that

+ o(l).

Annales de I’lnstituf Henri Poincar6 Probabilitk et Statistiques


Note that

G(x) = 2 ci exp (N(LZ, log Z)) i=l

Since for Oi = Oi (6) small enough and 15% E Oi one has that

l(L”,,log 2) - WMIP) - WPYPLj*))l I 6.

one sees that for 6 < min+j H(@ 1~,‘)/2 and all N large enough it holds that

IE(N~(l{L:EO.]~c% (gg))

- jvEcN) l{Lx,EO,}( N> ( LX log @/&)) 1 = O( 1) .

Hence, with pi(v) = F(v) - (v,log(d&/&)),

The proof that B N 5 c, i.e. of (5), now proceeds exactly as the prOOf of boundedness of BE) in Theorem 1.

Next, let pi (N) = p(N)(.IL$ E Oi) E Ml(SN). One has

ff(P,!N’I(~YN)

( dpcN’

IaN) l{LEEO,} 1% d(CL: > =

pL’N’(L~ E 0;)

NEcN)

= -1ogZN + ( l{Lx,@,}

( F(L$) + j$ log d;l,e;;E,

(xo >

p(N’(Lc E Oi)

- log/JN)(L$ E Oi) 1

= p’“‘(L$ E Oi) (o(l) + ~E(“)(l{L:,o*)mm - A(P))))

= O(l) > (18)

where the last two equalities follow from the convergence #dN)(LE E 0;) +N-+m c; > 0 and the same argument leading to the boundedness of Bg) in Theorem 1.

Vol. 35, no l-1999

98 G. BEN AROUS AND 0. ZElTOUNI

Let now

It follows from (3) and (18) that if k = k(N) = o(N) then

Recall that H(pN,“Ipk,*) 5 H(p(N)IpN;*) 5 C. Hence, dpA’;k/dpk.’ exists, and, denoting by [la - bll,,, the variation distance between two measures.

i=l

where the last limit is due to the inequality ](a - b(l,,, 5 ~/m and (19). One concludes that

and hence

Next, with f = log(dp”*“/dp”‘*) E Cb(Sk),

H(pN,klpk’*) = (f, pN,k) > (20)

A~zmzles de I’lnstitut Hrnri Poincurd ProbabilitCs et Statistiques


whereas, (at least for N/k integer, the general case following by truncation),

= SUP fECh(SN) (

(f,~‘“‘) - lo+?fdiL’.*)

Combining (20) and (21), one arrives at

as soon as k = o(N). 0

Proof of Theorem 3. - By [l, Page 115, (3.6)], one sees that the law LN,N is Gibbsian with respect to P, -‘N(pO) (with S = C( [0, T], lRd), and bounded, continuous potential F E i$Cb( S), s = T V 3, see [ 1, Page 118, Corollary 3.211). Hence, Theorem 3 follows from Theorem 1 as soon as one establishes that the variational problem associated with F possesses a unique, non degenerate minimum which is pi. But this is the content, respectively, of [ 1, Page 136, Corollary 3. lo] and [ 1, Page 122, Theorem 2.71. cl

4. THE DEGENERATE CASE

For simplicity, we assume (Al) in the degenerate case. Let d = dimK(,u*), let {ej};=l d eno e t an orthonormal base of K(p*) in Lz(S, CL*), and define X(t) by

X(t) = inf{I(v) - rn, : (ej, v) = (ej, p*) + tj , 1 I j < d} , t E IR” .

Vol. 35. no l-1999.


We refer to [l] (notably, the proof of Theorem C) for properties of X(t). In order to state our last hypothesis, we refer to the construction of

the embedding into the space B described in the proof of Theorem 1. Let {tij},j = l,..., d denote an orthonormal basis in the d-dimensional space of 4 E B* achieving equality in (1 1 ), such that $j(Th (I/ - /L*)) = (ej,V - /I*). Defining Q : B + IR. by q(x) = (&(z); . . .? dj(Z)), and A : B --+ JR by A(4) = inf{IJ(v]p) : Z’/,(U) = +}, it holds that (see 14, Pg. 1721, and use [l, Lemma 3.81 to identify x(t) of [l] with X(t) of 141)

X(C) = sup{F(z + Th(p,*)) - A(a: + Th(b*)) - mf1 : q(z) = t>.

In order to apply [4] (i.e., to obtain certain local limit theorems uniformly), we will need the following smoothness hypothesis on the finite dimensional measures ~1 0 q-l. (A2) The characteristic function of the measure 11, o q-l on Rd is in L,, some 00 > p 2 1.

THEOREM 4. - Assume (Al) and that pL* is degenerate. Further, assume (A2) and that x(t) -t+~ LjltlI*, some p > 3 integer and L > 0.

Then, l-q”) ) (p”)SN) = O(jp2/“).

As a consequence, for any k(N) = o(N’/“),

(22)

Proof of Theorem. - Throughout this proof, C denotes a constant whose value may change from line to line but which is always independent of N. We essentially follow the proof of Theorem 1, whose notations we adopt, except that one has to condition on the degenerate directions, as in 141. Here, with BN defined in (8), it is enough to prove that BN = O(N1-2/p). By [l], Theorem C, we know that ZNepNmO = O(N”(1/2-1/p)), and hence B$) = O(log(N)). Therefore, (12) is replaced by

BE) 5 CjjT-d(+-;) .I

jqF(~“,) - F(~*)I~N(F(LX”)-E’(~‘))~(~*)Q~N.

(23) By standard large deviations as in (13), one can localize the computation of (23) to any fixed neighborhood 0 of p*.

Letting now q(Th(Lc - p*)) = t(x), one may write Th(L$ - P*) = K(x) + W(x) with q(V,,,,) = t and q(W,(,)) = 0. Further,

Ann&s de /‘hstirut Henrr P&mm! ProbabilitCs et Statistiques


since IN(F(L”,) - F(p*)J 5 ClTh(m(Lg - CL*))& it holds that IN(F(L”,) - F(p*)( 5 CNIV&12 + CNIW,(,)l2,. Fix now c > 0 large enough such that [4, Proposition 3.121 can be applied, and take the neighborhood 0 to be such that Lg E 0 + IV,(,)1; + IW,(,)I$r < c-l. We now have

where the last inequality is due to the uniform CLT contained in [4, Proposition 3.121 (note that we are working in the non-degenerate directions, and follow the same computation as in Theorem 1 when deriving (17)). On the other hand, following the computation leading to [4, (4.2)-(4X$], one finds that

5 CNd/2 .I

(Nlt(2)e-LNltl’& 5 cNd(i-$)+=$ .

Combining the last inequality with (24) and (23) yields the desired estimate on BE) and hence the theorem. 0

Proof of Corollary 1. - We consider the case Q: = 0, the general case being similar. The argument leading from Case III to case I being exactly as outlined in the proof of Theorem 2, we deal with cases I and II only. Throughout this proof, C denotes a constant whose value may change from line to line but which is independent of N. Recall that

BN = N IE’(L;k;‘h(L~)wN

[ T lyN]

By theorem 2 of [o], one knows that 2,~ = C(l + o(1)) (case 1) while 2, = CN1i4( 1 + o(1)) ( case II) for some constant C (which may be computed, although we do not need this computation). Let YN = JE(LN,z). Then

BN = O(10g 2~) + /32,’ .I

Y,$e,‘y~dp@N

Vol. 35, Ilo l-1999


Next, by Chebycheff’s inequality, denoting A(6) = log E( eesl) = logcosh 0, one has

A direct computation reveals that A’(0) = R(“)(O) = 0, while i4”(O) = 1, and

WEi1 Y=

max A(“)(E) + A(4)(0) < o

5! 4! . Hence, in Case I, one gets

while in Case II one gets

‘2 E(YNe %,/2) 5 21” ,T2e-1-‘4/NdZ 5 ~~“14 I’y xze-Yz4& = CN”/a 0 . 0

One conclude that B,v = O(1) (Case I) and B,v = O(N’/2) (Case II). The conclusion of the corollary follows. 0

REFERENCES

[I] G. BEN AROUS and M. BRUNALJD, MPthode de Laplace: &de variationnelle desPuctuation.s de diffusions de type “champ moyen “, Stochastics, Vol. 31, 1990, pp. 79-144.

[2] G. BEN AROUS, Lecture notes, St. Flour, 1995. [3] E. BOLTHAUSEN, Laplace approximations for sums of independent random vectors, PTRF,

Vol. 72, 1986, pp. 305-318. [4] E. BOLTHAUSEN, Laplace approximations for sums of independent random vectors, part II:

degenerate maxima and manifold of muxima, Prob. Theory Rel. Fields. Vol. 76, 1987, pp. 167-206.

[?I] I. CSISZAR, Sanov property, generalized I-projection and a conditional limit theorem, Ann. Probab., Vol. 12, 1984, pp. 768-793.

[6] A. DEMBO and 0. ZEITOUNI, Refinements of the Gibbs conditioning principle, Prob. Theory Rel. Fields, Vol. 104, 1996, pp. I-14.

171 P. DIACONIS and D. FREEDMAN, Finite exchangeable sequences, Ann. Probab., Vol. 8, 1980, pp. 745-764.

[8] T. FUNAKI, A certain class of diffusion processes associated with nonlinear parabolic equations, Z. Wahr. vetw. Geb., Vol. 67, 1984, pp. 33 l-348.

[9] A. MARTIN-LBF, A Laplace approximation for sums of independent random variables, Z. Wahr. ver. Gebiete, Vol. 59, 1982, pp. 101-I 15.

[lo] F. PAPANGELOU, On the GaussianJluctuations $the critical Curie- Weiss model in statistical mechanics, Prob. Theory Rel. Fields, Vol. 83, 1989, pp. 265-278.

(Manuscript received November 17, 1996; Revised version received January 27, 1998 and June 16, 1998.)

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