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www.elsevier.com/locate/anihpb

Asymptotic behavior of the magnetization for the perceptron model

David Márquez-Carreras

a,1

, Carles Rovira

a,2

, Samy Tindel

b,3,

aFacultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, Spain bInstitut Élie Cartan, Université de Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy, France Received 8 December 2003; received in revised form 10 December 2004; accepted 15 April 2005

Available online 18 November 2005

Abstract

In this paper, we show that, in case of a perceptron model for which the number of outputs is a small proportion of the size of the system, the limiting behavior of the magnetization of a given spin, namely the random variableσk, can be identified. In fact, we prove aL2convergence for a collection of those random variables.

2005 Elsevier SAS. All rights reserved.

Résumé

Dans cet article, nous montrons que, dans le cas d’un modèle du perceptron pour lequel le nombre de sorties est une proportion suffisamment petite de la taille du système, on peut identifier le comportement limite de la magnétisation d’un spin quelconque, c’est-à-dire de la variable aléatoireσk. Nous montrons en fait une convergence dansL2pour une famille de telles variables.

2005 Elsevier SAS. All rights reserved.

MSC:60G15; 82D30

Keywords:Spin glasses; Perceptron model; Cavity method

1. Introduction

One of the basic problems in neural computation theory is the supervised learning for pattern recognition by a simple perceptron (we refer to [2] and references therein for a detailed account on that theory, from a physical and numerical point of view). This problem can be summarized in the following way: for two given integersN, M1 andi∈ {1, . . . , N}, the input of the perceptron isη=1, . . . , ηN)withηi∈ {−1,1}, and its output is of the form

Ok=u

iN

wi,kηi

, kM,

* Corresponding author.

E-mail addresses:marquez@mat.ub.es (D. Márquez-Carreras), carles.rovira@ub.edu (C. Rovira), tindel@iecn.u-nancy.fr (S. Tindel).

1 Partially supported by DGES grant BFM2000-0607.

2 Partially done while the author was visiting the Université Paris 13 with a CIRIT grant. Partially supported by DGES grant BFM2000-0607.

3 Partially written with a support of a fellowship grant at theCentre de Recerca Matemàtica(Bellaterra, Barcelona). The author would like to thank this institution for its warm hospitality.

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2005.04.005

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for a collection of real weights{wi,k; iN, kM}, and whereu:R→Rshould be, for numerical purposes, a step function. Note however that, in many case,uis taken as a regularization of the sign function, whose derivatives are allowed to grow at exponential speed withN. This is assumed for mathematical convenience, and also because it reproduces better the actual behavior of the brain synapses, from which this kind of model is inspired.

The supervised learning procedure consists then in providingp inputsη1, . . . , ηp, jointly withp desired output patternsζ1, . . . , ζp (called the training set), and, if possible, try to find the weightswi,kmatching the inputs and the outputs, i.e. such that

u

iN

wi,kηli

ζkl=0, kM, lp. (1)

Before computing those weights, a natural and classical question is to get some information about the theoretical capacity of our model, that is the maximal number of patterns for which the problem (1) can be solved. Without any a priori knowledge aboutwi,kandηl, the following assumptions can be done:

1. The inputsηare uniformly distributed inΣN= {−1,1}N.

2. The hyperplanes defined by the weightswi,k are also taken as “uniformly” distributed random planes, which is equivalent to modelwi,kas an i.i.d. family of standard Gaussian random variables, normalized byN1/2in order to get a term of order 1. That is, one chooses

wi,k= gi,k N1/2,

where{gi,k; i1, k1}is a family of independent standard Gaussian random variables.

However, it is a hard task to solve the system (1). Thus, in that context, one step further in the analysis is to replace it by the study of an associated Gibbs measure, a usual trick known as the finite temperature approximation. In our case, if one wishes to know on how many bitsMthe output can be coded, a natural measure to consider is the one whose density with respect to the uniform measureµNonΣN is given, for a typical configurationσ=1, . . . , σN)ΣN, byZN,M1 exp(−HN,M(σ )), with

HN,M(σ )=

kM

u

N1/2

iN

gi,kσi

, ZN,M=

σΣN

exp

HN,M(σ )

. (2)

Notice that, in order to get the formula (2), one has to remove all the constant weightsζkl, and to include the usual inverse of the temperature parameterβinto the functionu, which are also two standard tricks.

It is now easily seen that a phase diagram for the model given by (2), known as the perceptron model, would give some essential information about the capacity of the original model. This phase diagram is far from being complete from a mathematical point of view. However, the deep study initiated in [4, Chapter 3] (see also [3] in a slightly different context) leads to a rigorous replica symmetric solution whenMis a (small enough) proportion ofN, i.e.M= αN withαsmall enough. This replica symmetric solution indicates that the system does not behave too chaotically whenN→ ∞, and hence that the capacity of the perceptron is not reached yet.

Another way of looking at the regularity of the system whenN→ ∞ is to investigate the asymptotic behavior of the magnetization of an arbitrary family of spinsσ1, . . . , σk fork1, that is the limit of the laws of the random variables

σ1, . . . ,σk,

where·designates the average with respect to the measure defined by (2). In fact, in most engineering applications, the classical way to illustrate the self averaging phenomenon for this kind of model is to show that those quantities converge to some independent and identically distributed centered random variables that can be clearly identified, by analogy with the fact that the magnetization vanishes for the Ising model at high temperature. Therefore, our aim in this paper will be to give an answer to that natural problem, which can be summarized in the following theorem:

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Theorem 1.1.Assumeusatisfies the assumptions

(H1) There exist a constantD >0and a large enough numberLsuch that|u(x)|Dfor allx andLeLD1.

(H2) There exists a positive constant L and a small enough constant c3 such that, for any l 12, |u(l)| LeLDec3N.

Then, there exists a constantr0, that will be defined at Section2, such that, for any fixedm1,

im

E

σi −tanh(r1/2zi)2

κm

N1/2,

where κm is a positive constant, and(z1, . . . , zm)is a family of independent standard Gaussian random variables (which still depend, however, of the randomness contained ing).

Let us give now some hints about the way one can get that result: of course, our proof will be inspired by the methods introduced in [4], and particularly by the smart path method, which is an efficient way of interpolating two Gaussian vectors, once the limiting local fields corresponding to (2) can be guessed. In our case, if the natural path is quite easy to figure out (see (8) for a heuristic justification), the computations along that path are long and sometimes involved. In particular, the path (18), which seems to be the obvious one at first sight, lead us to a series of technical problems. Like in [1], we have chosen to give some details of our calculations for sake of readability, hoping that the final aspect of our paper will not be too scary.

Notice also that the optimal expected rate of convergence in our theorem would be inN1. We stopped atN1/2 and did not check the computations allowing to get this optimal rate, but we believe that it could be reached along the same lines we have developed in this paper. In any case, let us recall again that our result answers a natural question for the perceptron model, and corresponds to one of the basic facts that practitioners are looking for, as mentioned in [2]. Observe also that our way to derive Theorem 1.1 is certainly not the only one, and we hope that, in the future, some easier proofs can be elaborated. And it is worth pointing out that the methods contained in [3] could potentially yield one of these simplifications. However, the latter paper is based on some convexity assumptions on the functionu, and some spherical hypothesis on the spin model, which makes it hard to adapt to our situation.

Our paper is divided as follow: in the next section, we will introduce the basic notations concerning the perceptron model, and recall some results taken from [4]. Section 3 is devoted to state an important intermediate result (Proposi- tion 3.1) on our way to the proof of Theorem 1.1, to the definition of the Gaussian path we will use in the sequel, and to the first computations along this path. Some new Gibbs measures appear naturally in those computations. We will introduce them at Section 4. The main step towards Theorem 1.1 will then be achieved with an asymptotic expansion of a general term involving the measures introduced at Section 4. This will be done at Section 5. Then, we will apply those results in order to prove Proposition 3.1 at Section 6. Theorem 1.1 will be proved at Section 7. Eventually, let us mention that, in the remainder of the paper,κ, c, c1, . . .will stand for some positive constants, that can change from line to line.

2. Preliminary results

As mentioned in the introduction, the state space of our model will beΣN= {−1,1}N for a givenN∈N, and settingµNfor the uniform measure onΣN, we will consider the random measure whose density with respect toµNis given byZN,M1 exp(−HN,M(σ )), whereZN,Mhas been introduced at relation (2), and where we recall thatHN,M(σ ) is defined by

HN,M(σ )=

kM

u

N1/2

iN

gi,kσi

.

In the above formula,M stands for a positive integer such thatM=αN for a givenα >0,uwill be a continuous function defined onRsatisfying conditions (H1) and (H2) given in the introduction, and{gi,k; i1, k1}is a fam- ily of independent standard Gaussian random variables. Note thatuandαwill depend onN, though this dependence will be kept implicit for sake of readability.

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We will first give an account on the results obtained in [4] concerning the perceptron model. Those results are based on the smart path method, that is a way to single out continuously the last spinσN from the other ones in the definition ofHN,M. This smart path can be described as follows: fort∈ [0,1],σΣN, set

Sk,t(σ )= 1

N

1/2

iN1

gi,kσi+ t

N 1/2

gN,kσN, (3)

and consider the Hamiltonian

HN,M,t(σ )=

kM

u Sk,t(σ )

+σN

r(1t )1/2

Y, (4)

whereris a positive coefficient whose exact value will be given later on, andYis a standard Gaussian random variable independent of all the other randomness inHN,M. For a given functionf:ΣNn →R, set nowftfor the Gibbs mean off with respect to the HamiltonianHN,M,t, and

νt(f )=E ft

.

Then, on one hand, some simplifications in the computation ofνt(f )can be done whent=0:

Lemma 2.1.Letfbe a function defined onΣNn1, andI be a subset of{1, . . . , n}of cardinality|I|. Then ν0

f

lI

σNl

=E

tanh(r1/2Y )|I| ν0(f).

Notice that, in the previous result,σ1, . . . , σnare understood asnindependent configurations underGN, usually called replicas ofσ.

On the other hand, the derivative ofνt(f )with respect to the parametertcan be computed explicitly.

Proposition 2.2.Suppose thatuis twice differentiable. Writew=u+(u)2. Then, for0< t <1andf:ΣNn →R, we have

νt(f )= 3 j=1

Kj, where

K1=α 2

ln

νt

w(SM,tl )f

α 2t

w(SM,tn+1)f , K2is defined by

K2=α

1l<ln

νt

σNlσNlu(SM,tl )u(SM,tl )f

αn

ln

νt

σNlσNn+1u(SM,tl )u(SM,tn+1)f

+αn(n+1) 2 νt

σNn+1σNn+2u(SM,tn+1)u(SM,tn+2)f , and

K3= −r

1l<ln

νtNlσNlf )tNlσNn+1f )+n(n+1)

2 νtNn+1σNn+2f )

.

The drawback of Proposition 2.2 is that some terms involvinguanduappear in the formula, while one would like to have only a mild control on those quantities. However, using some integration by parts arguments, some asymptotic results for the perceptron model are obtained in [4]: first of all, define, for two replicasσl, σlofσ, the quantity

Rl,l= 1 N

iN

σilσil.

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This quantity, called the overlap ofσlandσl, will be, as usual in the spin glass theory, a central object in the study of the model. It is then essential to obtain some sharp results on the asymptotic behavior ofR1,2, that we will describe now: letq andr(notice thatrhas already been introduced in (4)) be the solution to the system

q=E

tanh2(r1/2Z)

, r=αE

Λ2

q1/2Z, (1q)1/2

, (5)

whereΛis defined onR2by

Λ(x, y):=E[ξexp(u(x+ξy))]

yE[exp(u(x+ξy))]=E[u(x+ξy)exp(u(x+ξy))] E[exp(u(x+ξy))] ,

and where the second equality is obtained by a Gaussian integration by parts. In the preceding formulae, Z andξ stand for two independent standard Gaussian random variables. Observe that, under hypothesis (H1), the system (5) admits a unique solution. Then the following inequalities hold true (notice that these inequalities are shown in [4]

under slightly weaker assumptions):

Proposition 2.3.Under hypothesis(H1)and(H2), we have, for anyk0, ν

(R1,2q)2k

Lk N

k

. Furthermore, we have

ν(R1,2q) L

N. (6)

Notice that the computations leading to this exponential self-averaging property for the overlap also yield the replica symmetric solution for the model. We refer to [4] for this last formula.

Let us end this section by recalling an elementary integration by parts formula we will use throughout the paper:

forp1, let(g1, . . . , gp)be a Gaussian vector inRp, andF:Rp→Rbe aC1function having at most exponential growth together with its gradient. Then

E

g1F (g1, . . . , gm)

=

lm

E[g1gl]E

xlF (g1, . . . , gm)

. (7)

3. Definition of the path

A first step in order to prove Theorem 1.1 is to achieve an asymptotic decomposition ofσNseparating, on one hand, the randomness contained in{gN,k; kM}, and on the other hand, some terms involving{gi,k; iN−1, kM}.

First of all let us introduce a new Hamiltonian

HN1,M(σ )=

kM

u

N1/2

iN1

gi,kσi

=

kM

u(Sk,0),

where the quantitiesSk,thave been defined at (3), and let·denote the average with respect to the Gibbs measure induced byHN1,M.

Then, the following heuristic steps can be performed: we have

HN,M(σ )HN1,M(σ )+N1/2

kM

u(Sk,0)gN,kσN, and hence

σN sinh((1/N1/2)

kMgN,ku(Sk,0))

cosh((1/N1/2)

kMgN,ku(Sk,0)). (8)

We will show in the sequel that this crude estimate holds true in the limitN→ ∞, and in fact, using a direct analogy with the self averaging properties of the SK model, we will prove the following result:

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Proposition 3.1.Under hypothesis(H1)and(H2), there exists a strictly positive constantκsuch that E

σN −tanh 1

N1/2

kM

gN,k

u(Sk,0)

2

κ

N1/2. (9)

Most of the remainder of the paper will be devoted to prove (9), and let us begin here with some simple considera- tions, that is, the definition of the Gaussian path we will use: fort∈ [0,1], let us consider the function

ϕ(t )=E

σNt−tanhYt2 , where

Yt= t

N

1/2

kM

gN,k

u(Sk,0)

+

r(1t )1/2

Y.

Notice then thatϕ(1)is the left-hand side of (9), and on the other hand, using thatσN0=tanh(r1/2Y ),it is easy to check thatϕ(0)=0. By means of the elementary equation

ϕ(1)ϕ(0)= 1 0

ϕ(t )dt,

we can see that (9) will be achieved as soon as we can show that ϕ(t ) κ

N1/2,

for anyt(0,1). Hence, our first task will be to compute the derivative ofϕ. In fact, we will start by decomposingϕ into three terms:

ϕ(t )=ϕ1(t )+ϕ2(t )+ϕ3(t ), (10)

with

ϕ1(t )=E σN2t

, ϕ2(t )=E

tanh2(Yt) , ϕ3(t )= −2E

σNttanhYt .

Then, we will compute the derivatives ofϕ1,ϕ2andϕ3separately.

Using two replicas ofσ, we getϕ1(t )=νtN1σN2).Then, as a direct consequence of Proposition 2.2, we get the following lemma.

Lemma 3.2.For anyt∈ [0,1], we have ϕ1(t )=

k6

A1,k(t ), with

A1,1(t )=ανt

u+(u)2

(SM,t1 N1σN2 , A1,2(t )= −ανt

u+(u)2

(S3M,tN1σN2 , A1,3(t )=ανt

u(SM,t1 )u(SM,t2 ) , A1,4(t )= −4ανt

u(SM,t1 )u(SM,t3 N2σN3 , A1,5(t )=3ανt

u(SM,t3 )u(SM,t4 N1σN2σN3σN4 , A1,6(t )= −r

1−4νtN1σN2)+3νtN1σN2σN3σN4) .

Differentiating the functionϕ2(t )and using the integration by parts formula (7), we also obtain the following result.

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Lemma 3.3.For anyt∈ [0,1], we have ϕ2(t )=A2,1(t )+A2,2(t ), with

A2,1(t )=αE

1−tanh2(Yt)

1−3 tanh2(Yt)

u(SM,0)2

, A2,2(t )= −rE

1−tanh2(Yt)

1−3 tanh2(Yt) .

Now, differentiatingϕ3(t )and using again the integrate by parts formula (7) we get the next lemma.

Lemma 3.4.For anyt∈ [0,1], we have ϕ3(t )= −2

i4

A3,1,1,i(t )

i5

A3,1,2,i(t )+A3,2,1(t )A3,2,2(t )

, with

A3,1,1,1(t )=α 2E

σN

u(2)(SM,t)+

u(SM,t)2

ttanhYt , A3,1,1,2(t )= −α

2E

σN2u(SM,t1 )u(S2M,t)

ttanhYt

,

A3,1,1,3(t )=α 2E

u(SM,t)

t

u(SM,0)

1−tanh2(Yt) , A3,1,1,4(t )= −r

2E

1−tanh2(Yt) , the termsA3,1,2,i(t )being defined by

A3,1,2,1(t )=α 2E

σN1

u(2)(SM,t2 )+

u(S2M,t)2

ttanhYt , A3,1,2,2(t )=α

2E

σN1u(SM,t1 )u(SM,t2 )

ttanhYt , A3,1,2,3(t )= −αE

σN1σN2σN3u(SM,t1 )u(SM,t2 )

ttanhYt

, A3,1,2,4(t )=α

2E

σN1σN2u(SM,t2 )

t

u(SM,0)

(1−tanh2Yt) , A3,1,2,5(t )= −rE

σNttanhYt +rE

σN1σN2σN3ttanhYt

r 2E

σN1σN2t(1−tanh2Yt) . The termsA3,2,1(t )andA3,2,2(t )are respectively obtained by

A3,2,1(t )=α 2E

u(SM,t)

t

u(SM,0)

1−tanh2(Yt)

α 2E

σN1σN2u(SM,t1 )

t

u(SM,0)

1−tanh2(Yt)

αE σN

tu(SM,0)2tanhYt

1−tanh2(Yt) , and

A3,2,2(t )=r 2E

1−tanh2(Yt)

r 2E

σN1σN2t

1−tanh2(Yt)

rE

σNttanhYt

1−tanh2(Yt) . 4. Introducing a new measure

All the preceding considerations show that, in order to prove relation (9), we should be able to expand and control any term of the form

f (σ1, . . . , σn1)U1(SM,t1 , . . . , SM,tn1 )

t

U2(SM,01 , . . . , SM,0n2 )

, (11)

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for n1, n2∈N, U1:Rn1 →R, U2:Rn2 →R andt(0,1]. In this section, we will begin to develop some new tools that should allow us to handle those terms. More precisely, we will introduce some new Gibbs measures in the following way: first of all, fort∈ [0,1],σΣN, set

HN,M,t (σ )=

kM

u Sk,t(σ )

+σNbt,

where we recall thatY is a standard Gaussian random variable independent of the remaining randomness contained inHN,M,t , and

bt=

r(1t )1/2

Y, bt=bt1(0,1](t ). (12)

Write thenGt for the Gibbs measure related to this Hamiltonian, and notice that HN1,M(σ )= −

kM

u

Sk,0(σ )

=HN,M,0 (σ ).

For a fixed(t1, . . . , tn)∈ [0,1]n,let us consider now a new Gibbs measureGt

1,...,tnonΣNn, defined by its average on functionsf:ΣN→R:

f(t1,...,tn):=

σ1,...,σnfexp(

ln(

kM1u(Sk,tl

l)+σNlbt

l))

σ1,...,σnexp(

ln(

kM1u(Sk,tl

l)+σNlbtl)) , wheref =f (σ1, . . . , σn).

Now, the terms of the form (11) can be expressed using a Gibbs measure of the formG: let us consider the notation n:=n1+n2, as well as

U1:=U1(SM,t1 , . . . , SM,tn1 ), U2:=U2(S1M,0, . . . , SM,0n2 )

andf =f (σ1, . . . , σn1),U2:=U2(SM,0n1+1, . . . , SM,0n ).Then, fort∈ [0,1], f U1tU2=

σ1,...,σn1f U1exp(

ln1[

kMu(Sk,tl )+σNl(r(1t ))1/2Y])

σ1,...,σn1exp(

ln1[

kMu(Sk,tl )+σNl (r(1t ))1/2Y])

×

σ1,...,σn2U2exp(

ln2[

kMu(Sk,0l )])

σ1,...,σn2exp(

ln2[

kMu(Sk,0l )]) , and using the relation

HN,M,t (σ )= −HN,M 1,tl)+u(SM,t), we get, fort∈ [0,1],

f U1tU2=

σ1,...,σnf U1U2exp(−

ln1HN,M,t l)

n1<lnHN,M,0 l))

σ1,...,σnexp(−

ln1HN,M,t l)

n1<lnHN,M,0 l))

=f U1U2exp(

ln1u(SM,tl )+

n1<lnu(SlM,0))(t,...,t,0,...,0)

exp(

ln1u(SM,tl )+

n1<lnu(SM,0l ))(t,...,t,0,...,0)

(13) where in(t, . . . , t,0, . . . ,0)we haven1timestandn2times 0.

On the other hand, the measuresGt

1,...,tnshare some of the good properties of the original Gibbs measure, starting from the one concerningwidely spreadsequences of Gaussian random variables, a crucial notion that we recall briefly here (see however [4] for a detailed account on that topic).

Definition 4.1.Forn1, a centered Gaussian family(X1, . . . , Xn)is said to be widely spread if for eachlnwe can write

Xl=Xl+

k =l

cl,kXk,

(9)

where thecl,kare some real coefficients,Xlis independent of the random variablesXk fork =l, and E(Xl)2

κ1,

for a fixed positive constantκ1.

Then, as in Lemma 3.3.3 of [4] we can prove that the measuresGt

1,...,tn satisfy:

Lemma 4.2.Assumeusatisfies(H1)and(H2), fix(t1, . . . , tn)∈ [0,1]n,and set, for anyt∈ [0,1]andσΣN, gt(σ )= 1

N1/2

iN1

giσi+ t

N 1/2

gNσN,

where {gi; iN}is a sequence of independent standard Gaussian random variables, also independent from the randomness contained inGt

1,...,tn. Then, there exists some constantsκ, c1>0such that Gt

1,...,tn

1, . . . , σn)ΣNn;gt11), . . . , gtnn)is not widely spread

κnexp(−c1N ).

Let us recall that the constantq∈ [0,1]has been introduced in (5) and define, forln, the quantity

θl=q1/2W+(1q)1/2ξl, (14)

whereW and{ξl; l1}are independent standard Gaussian random variables. For a fixedt∈ [0,1], consider also the variables

gv,tl =v1/2SM,tl +(1v)1/2θl, (15)

for anyv∈ [0,1]. In the sequel, we will denote byEξthe partial expectation with respect to the randomness contained in{ξl; l1}. With Lemma 4.2 in mind, one can get the following proposition

Proposition 4.3.Assumeusatisfies(H1)and(H2), and fix(t1, . . . , tn)∈ [0,1]n. Then we have 1. The overlapR1,2self-averages intoq:

E

(R1,2q)2

(t1,t2)

κ

N. (16)

2. Consider some integersn1,s1, . . . , snandr1, . . . , rn∈ {0,1}such that, settings=

jnsj andr=

jnrj, we haves, r6. Set

V (x1, . . . , xn)=exp

ln

u(xl)

, U (x1, . . . , xn)=ss

1,...,snV (x1, . . . , xn), P (x1, . . . , xn)=

jn

u(xl)rj

.

Then, forf:ΣNn →Rsuch that|f|1, we have, for anyv∈ [0,1], fEξ[U (g1v,t

1, . . . , gv,tn

n)]P (SM11,t

1, . . . , SMn1,t

n)(t1,...,tn)

Eξ[V (g1v,t

1, . . . , gv,tn

n)](t1,...,tn)

κ e2nD

|f|

+ec2N

, (17)

for some positive constantsκ, c2.

Proof. This proof is a long and easy elaboration of the computations performed at [4, Sections 3.3 and 3.4], that we will not include here. However, notice that we have to assume here that the derivatives ofuare (midely) bounded up to order 12. This is due to the presence of bothU (gv,t1

1, . . . , gv,tn

n)andP (S 1M1,t

1, . . . , SMn1,t

n)in the left-hand side of (17). In order to treat those terms, we will separate them using Schwarz’s inequality, that will bring out up to 12 derivatives ofu. 2

(10)

5. An asymptotic expansion

In this section, we will develop the tools that we will need in order to get an expansion of the terms of the form (11), which is a key step in the proof of Theorem 1.1. In fact, once the averagesf(t1,...,tn) are defined, we will try, for a fixed(t1, . . . , tn), to find a natural path that will allow us to expand any term of the form (11), asymptotically inN.

Recall first thatSk,t is defined by (3),gv,t by (15),θl by (14),bt andbt by (12). Observe that, using the notation introduced at Section 4, we have

Yt= t

N

1/2

kM

gN,ku(Sk,0)expu(SM,0)(0)

expu(SM,0)(0) +bt.

We can turn now to the definition of the path we will use in order to get our expansion: forv∈ [0,1], set Yv=

t N

1/2

kM1

gN,k

Eξu(Sk,0)expu(gv,0)(0)

Eξexpu(gv,0)(0) +bt. (18)

In particular it is easy to check that Y0=

t N

1/2

kM1

gN,k

u(Sk,0)

(0)+bt, Y1=Yt

t N

1/2

gN,Mu(SM,0)expu(SM,0)(0)

expu(SM,0)(0)

. (19)

Using this auxiliary path, we will get the announced expansion as follows:

Proposition 5.1.Letube a function satisfying(H1)and(H2). Consider an integerm, nonnegative integerss1, . . . , sm withs=

lmsl2. Set also V (x1, . . . , xm)=exp

lm

u(xl),

U (x1, . . . , xm)= sV

∂x1s1· · ·∂xmsm.

Letf be a function onΣNmsuch that|f|1andΨ a bounded function with bounded derivatives of any order. Then, if we define, fortl∈ {0, t}, lm,

Q=E

f U (SM,t1

1, . . . , SmM,t

m)(t1,...,tm)

V (S1M,t

1, . . . , SM,tm

m)(t1,...,tm)

Ψ (Yt)

, we have

Q=E

f(t1,...,tm)Ψ (Y0) E

U (θ1, . . . , θm) EξV (θ1, . . . , θm)

+O(N1/2).

In particular, we have E

f(t1,...,tm)Ψ (Yt)

=E

f(t1,...,tm)Ψ (Y0)

+O(N1/2).

The proof of this proposition will we based on a technical lemma that we will present now. First of all, let us label some basic but useful results that we will use further on: forl, l1, t∈ [0,1], we have

E

(SM,tl )2

=1+t−1

N , (20)

E[SM,tl SM,tl ] :=t,tl,l=Rl,l+t t−1

N σNl σNl. (21)

Références

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