A functional central limit theorem for interacting particle systems on transitive graphs

(1)

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A functional central limit theorem for interacting

particle systems on transitive graphs

Paul Doukhan, Gabriel Lang, Sana Louhichi, Bernard Ycart

To cite this version:

Paul Doukhan, Gabriel Lang, Sana Louhichi, Bernard Ycart. A functional central limit theorem for

interacting particle systems on transitive graphs. Markov Processes And Related Fields, Polymat

Publishing Company, 2008, 14 (1), pp.79-114. �hal-00268278�

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arXiv:math-ph/0509041v3 12 Apr 2007

parti le systems on transitive graphs

P. Doukhan∗ , G. Lang† , S. Louhi hi,‡ and B. Y art§ February 7, 2008 Abstra t

A nite range intera ting parti le system on a transitive graph is onsidered. Assuming that the dynami s and the initial measure are invariant, the normalized empiri aldistributionpro ess onvergesindistributiontoa entereddiusionpro ess. As an appli ation, a entral limit theorem for ertain hitting times, interpreted as failuretimes ofa oherent systeminreliability,is derived.

Key words : Intera ting parti lesystem, fun tional entral limittheorem, hittingtime.

AMS Subje t Classi ation : 60K35, 60F17

∗

LS-CREST,URACNRS 220,Paris. &SAMOS-MATISSE-CES(StatistiqueAppliquéeet MOdélisa-tionSto hastique) Centred'E onomiedelaSorbonne,UniversitéParis1-Panthéon-Sorbonne. 90Rue de Tolbia ,75634ParisCedex 13,Fran e. doukhanensae.fr

†

AgroParisTe h, UMR MIA 518, INRA AgroParisTe h, 75005 Paris Fran e. gabriel.langagropariste h.fr

‡

Correspondingauthor: S.Louhi hi,UniversitédeParis-Sud,Probabilités,statistiqueetmodélisation, Bât. 425,91405OrsayCedex,Fran e. sana.louhi himath.u-psud.fr

§

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Intera ting parti le systems have attra ted a lot of attention be ause of their versatile modelling power (see for instan e [?, ?℄). However, most available results deal with their asymptoti behavior, and relatively few theorems des ribe their transient regime. In parti ular, entrallimittheorems for random elds have been available fora long time [?, ?, ?,?, ?, ?, ?℄, diusionapproximationsand invarian eprin iples haveaneven longer history ([?℄ and referen es therein), but those fun tional entral limit theorems that de-s ribethetransientbehaviorofanintera tingparti lesystemareusuallymu hlessgeneral than their xed-time ounterparts. Existing results (see [?, ?, ?, ?℄) require rather strin-gent hypotheses: spin ip dynami s on

Z

, reversibility, exponential ergodi ity, stationar-ity...(see Holleyand Strook'sdis ussion inthe introdu tionof[?℄). The mainobje tiveof this arti le is to prove a fun tional entral limit theorem for intera ting parti lesystems, underverymildhypotheses, usingsomenewte hniquesofweaklydependentrandomelds. Our basi referen e onintera ting parti le systems is the textbook by Liggett [?℄, and weshall trytokeep our notationsas lose tohisaspossible:

S

denotes the( ountable)set of sites,

W

the (nite)set ofstates,

X = W

S

theset of ongurations, and

{η

t

, t

≥ 0}

an intera ting parti le system, i.e. a Feller pro ess with values in

X

. If

R

is a nite subset of

S

,an empiri alpro ess isdened by ounting how many sites of

R

are inea hpossible stateattime

t

. This empiri alpro esswillbedenoted by

N

R

₌

_{N

R

t

, t

≥ 0}

,and dened as follows.

N

_t

R

= (N

_t

R

(w))

w∈W

,

N

t

R

(w) =

X

x∈R

I

_w

_(η

_t

_{(x)) ,}

where

I

w

denotes the indi ator fun tion of state

w

. Thus

N

R

t

is a

N

W

-valued sto hasti pro ess, whi h is not Markovian in general. Our goal is to show that, under suitable hypotheses,aproperlys aledversionof

N

R

onvergestoaGaussianpro essas

R

in reases to

S

. The hypotheses will be pre ised in se tions 2 and 3 and the main result (Theorem 4.1) willbestated andproved inse tion4. Here isa loose des riptionof our assumptions. Dealing with a sum of random variables, two hypotheses an be made for a entral limit theorem: weak dependen e and identi al distributions.

1. Weak dependen e: In order to give it a sense, one has to dene a distan e between sites, and therefore a graph stru ture. We shall rst suppose that this (undire ted) graph stru ture has bounded degree. We shall assume alsonite range intera tions: the onguration an simultaneously hange only ona bounded set of sites, and its value atone site aninuen e transitionrates onlyuptoaxed distan e (Denition 3.2). Then if

f

and

g

are two fun tions whose dependen e on the oordinates de- reasesexponentiallyfastwiththedistan efromtwodistantnitesets

R

1

and

R

2

,we shall prove that the ovarian ebetween

f (η

s

)

and

g(ζ

t

)

de ays exponentially fast in thedistan e between

R

1

and

R

2

(Proposition3.3). The entrallimittheorem 4.1will a tuallybeproved inamu hnarrowersetting,thatofgroupinvariantdynami sona transitivegraph(Denition3.4). However webelievethata ovarian einequality for generalniterangeintera tingparti lesystems isof independentinterest. Of ourse

(4)

2. Identi aldistributions: Inordertoensurethattheindi atorpro esses

{I

w

(η

t

(x)) , t

≥

0 }

are identi ally distributed, we shall assume that the set of sites

S

is endowed with a transitive graph stru ture (see [?℄ as a general referen e), and that both the transitionrates andthe initialdistributionare invariantbythe automorphismgroup a tion. This generalizes the notion of translation invarian e, usually onsidered in

Z

d

([?℄ p.36), and an be appliedtonon-latti e graphssu h astrees. Several re ent arti les have shown the interest of studying random pro esses on graph stru tures more generalthan

Z

d

latti es: see e.g. [?, ?, ?℄, and for generalreferen es [?, ?℄. Among the potential appli ations of our result, we hose to fo us on the hitting time of a pres ribed level by a linear ombination of the empiri al pro ess. In [?℄, su h hitting timeswere onsideredinthe appli ation ontextof reliability. Indeedthe sitesin

R

an be viewed as omponents of a oherent system and their states as degradation levels. Then a linear ombination of the empiri al pro ess is interpreted as the global degradation of the system, and by Theorem 4.1, it is asymptoti ally distributed as a diusion pro ess if the number of omponents is large. An upper bound for the degradation level an be pres ribed: the system is working as soon as the degradation is lower, and fails at the hitting time. More pre isely, let

f : w

7→ f(w)

be a mapping from

W

to

R

. The total degradationis the real-valued pro ess

D

R

₌

_{D

R

t

, t

≥ 0}

, dened by:

D

_t

R

=

X

w∈W

f (w)N

_t

R

(w).

If

a

is the pres ribed level, the failure time of the system will be dened as the random variable

T

_a

R

= inf

{t ≥ 0 , D

R

t

≥ a }.

Under suitable hypotheses, we shall prove that

T

R

a

onverges weakly to a normal distri-bution, thus extending Theorem 1.1 of [?℄ to systems with dependent omponents. In reliability (see [?℄ for a general referen e), omponents of a oherent system are usually onsideredasindependent. Thereasonseems tobemathemati al onvenien e ratherthan realisti modelling. Models withdependent omponentshavebeen proposed inthe setting ofsto hasti Petrinets[?,?℄. ObservingthataMarkovianPetrinet analsobeinterpreted asanintera tingparti lesystem, webelievethatthemodelstudiedhereisversatileenough tobeused inpra ti alappli ations.

The paper isorganized as follows. Some basi fa tsabout intera ting parti lesystems are rst re alled in se tion 2. They are essentially those of se tions I.3 and I.4 of [?℄, summarized here for sake of ompleteness, and in order to x notations. The ovarian e inequality for nite range intera tions and lo al fun tions will be given in se tion 3. Our main result, Theorem 4.1, will be stated inse tion 4. Some examplesof transitive graphs are proposed in se tion 5. The appli ation to hitting times and their reliability interpre-tation is the obje t of se tion 6. In the proof of Theorem 4.1, we need a spatial CLT for anintera ting parti lesystem atxed time,i.e. a randomeld. Wethoughtinteresting to

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Bolthausen[?℄on

Z

d

,butitusesasomewhatdierentte hnique. Allproofsarepostponed tose tion 8.

2 Main notations and assumptions

In order to x notations, we briey re all the basi onstru tion of general intera ting parti lesystems, des ribed inse tions I.3 and I.4 of Liggett'sbook [?℄.

Let

S

be a ountable set of sites,

W

a nite set of states, and

X = W

S

the set of ongurations,endowedwithitsprodu ttopology,thatmakesita ompa tset. Onedenes a Feller pro ess on

X

by spe ifying the lo al transition rates: to a onguration

η

and a nite set ofsites

T

isasso iateda nonnegativemeasure

c

T

(η,

·)

on

W

T

. Looselyspeaking, we want the ongurationto hange on

T

afteran exponential time with parameter

c

T,η

=

X

ζ∈W

T

c

T

(η, ζ).

Afterthat time, the ongurationbe omesequalto

ζ

on

T

, withprobability

c

T

(η, ζ)/c

T,η

. Let

η

ζ

denote the new onguration, whi h is equal to

ζ

on

T

, and to

η

outside

T

. The innitesimal generator shouldbe:

Ωf (η) =

X

T ⊂S

X

ζ∈W

T

c

T

(η, ζ)(f (η

ζ

)

− f(η)).

(1)

For

Ω

togenerate a Feller semigroup a tingon ontinuous fun tionsfrom

X

into

R

,some hypotheseshave tobeimposed on the transitionrates

c

T

(η,

·)

.

The rst ondition is that the mapping

η

7→ c

T

(η,

·)

should be ontinuous (and thus bounded, sin e

X

is ompa t). Let usdenote by

c

T

itssupremum norm.

c

T

= sup

η∈X

c

T,η

.

It is the maximal rate of hangeof a onguration on

T

. One essential hypothesis isthat the maximal rate of hange of a ongurationatone given site isbounded.

B = sup

x∈ S

X

T ∋ x

c

T

<

∞.

(2)

If

f

isa ontinuous fun tion on

X

,one denes

∆

f

(x)

asthe degree of dependen e of

f

on

x

:

∆

f

(x) = sup

{ |f(η) − f(ζ)| , η, ζ ∈ X

and

η(y) = ζ(y)

∀ y 6= x }.

Sin e

f

is ontinuous,

∆

f

(x)

tends to

0

as

x

tends to innity, and

f

is said to be smooth if

∆

f

issummable:

|||f||| =

X

x ∈ S

(6)

It anbeproved thatif

f

issmooth, then

Ωf

denedby(1)isindeeda ontinuousfun tion on

X

and moreover:

kΩfk ≤ B|||f|||.

Wealsoneedto ontrolthedependen e ofthetransitionratesonthe ongurationatother sites. If

y

∈ S

is a site, and

T

⊂ S

is a niteset of sites, one denes

c

T

(y) = sup

{ kc

T

(η

1 ,

· ) − c

T

(η

2 ,

· )k

tv

, η

1 (z) = η

2 (z)

∀ z 6= y },

where

k · k

tv

is the total variation norm:

kc

T

(η

1 ,

· ) − c

T

(η

2 ,

· )k

tv

=

1

2 X

ζ∈W

T

|c

T

(η

1 , ζ)

− c

T

(η

2 , ζ)

|.

If

x

and

y

are two sites su h that

x

6= y

,the inuen e of

y

on

x

isdened as:

γ(x, y) =

X

T ∋ x

c

T

(y).

Wewillset

γ(x, x) = 0

forall

x

. The inuen es

γ(x, y)

are assumed tobesummable:

M = sup

x∈ S

X

y∈ S

γ(x, y) <

∞.

(3)

Under both hypotheses (2) and (3), it an be proved that the losure of

Ω

generates a Feller semigroup

{S

t

, t

≥ 0}

(Theorem 3.9p.27of[?℄). Ageneri pro ess withsemigroup

{S

t

, t

≥ 0}

will be denoted by

{η

t

, t

≥ 0}

. Expe tations relative to its distribution, starting from

η

0 = η

will be denoted by

E

η

. For ea h ontinuous fun tion

f

,one has:

S

t

f (η) = E

η

[f (η

t

)] = E[f (η

t

)

| η

0 = η].

Assume now that

W

is ordered, (say

W =

{1, . . . , n}

). Let

M

denote the lass of all ontinuous fun tions on

X

whi h are monotone in the sense that

f (η)

≤ f(ξ)

whenever

η

_{≤ ξ}

. Asitwasnoti edbyLiggett(1985)itisessentialtotakeadvantageofmonotoni ity in order to prove limit theorems for parti le systems. The following theorems dis uss a number of ideas relatedto monotoni ity.

Theorem 2.1 (Theorem 2.2 Liggett, (1985)) Suppose

η

t

isaFellerpro esson

X

with semigroup

S(t)

. The following statement are equivalent :

(a)

f

∈ M

implies

S(t)f

∈ M

, for all

t

≥ 0

(b)

µ

1 ≤ µ

2

implies

µ

1 S(t)

≤ µ

2 S(t)

for all

t

≥ 0

.

Re allthat

µ

1 ≤ µ

2

provided that

R fdµ

1 ≤

R fdµ

2

for any

f

∈ M

.

Denition 2.2 A Feller pro ess is said to be monotone (or attra tive) if the equivalent onditions of Theorem2.1 are satised.

(7)

Theorem 2.3 (Theorem 2.14 Liggett, (1985)) Suppose that

S(t)

and

Ω

are respe -tively the semigroup and the generator of a monotone Feller pro ess on

X

. Assume further that

Ω

isa bounded operator. Then the followingtwo statements are equivalent:

(a)

Ωf g

≥ fΩg + gΩf

, for all

f

,

g

∈ M

(b)

µS(t)

has positive orrelations whenever

µ

does.

Re allthat

µ

has positive orrelationif

R fgdµ ≥ R fdµ R gdµ

for any

f, g

∈ M

. Thefollowing orollarygives onditionsunderwhi hthepositive orrelationproperty on-tinue to hold atlater times if it holds initially.

Corollary 2.4 [Corollary 2.21 Liggett,(1985)℄ Suppose thatthe assumptions of Theorem 2.3 are satised and that the equivalent onditions of Theorem 2.3 hold. Let

η

t

be the orresponding pro ess, where the distribution of

η

0

has positive orrelations. Then for

t

1 < t

2 <

· · · < t

n

the joint distribution of

(η

t

1 ,

· · · , η

t

n

)

, whi h isa probability measure on

X

n

, has positive orrelations.

3 Covarian e inequality

This se tion is devoted to the ovarian e of

f (η

s

)

and

g(η

t

)

for a nite range intera ting parti lesystem when the underlyinggraphstru ture has bounded degree. Proposition3.3 shows that if

f

and

g

aremainlylo atedon twonite sets

R

1

and

R

2

,then the ovarian e of

f

and

g

de ays exponentiallyin the distan e between

R

1

and

R

2

.

From now on, we assume that the set of sites

S

is endowed with anundire ted graph stru ture, and we denote by

d

the natural distan e on the graph. We will assume not only that the graph is lo ally nite, but also that the degree of ea h vertex is uniformly bounded.

∀x ∈ S , |{y ∈ S , d(x, y) = 1}| ≤ r ,

where

| · |

denotes the ardinality of a nite set. Thus the size of the sphere or ballwith enter

x

andradius

n

is uniformlybounded in

x

,and in reases atmostgeometri allyin

n

.

|{y ∈ S , d(x, y) = n}| ≤

r

_{− 1}

(r

−1)

n

and

|{y ∈ S , d(x, y) ≤ n}| ≤

r

_{− 2}

(r

−1)

n

_.

Let

R

be a nite subset of

S

. We shall use the followingupper bounds for the numberof verti es atdistan e

n

, oratmost

n

from

R

.

|{x ∈ S , d(x, R) = n}| ≤ |{y ∈ S , d(x, R) ≤ n}| ≤ 2|R|e

nρ

,

(4)

with

ρ = log(r

− 1)

.

In the ase of an amenable graph (e.g. alatti e on

Z

d

), the ballsizes have a subexpo-nentialgrowth. Therefore, for all

ε > 0

, there exists

c

su h that :

(8)

repla ing

ρ

by

ε

,for any

ε > 0

.

We are going to deal with smooth fun tions, depending weakly on oordinates away from a xed nite set

R

. Indeed, it is not su ient to onsider fun tions depending only on oordinates in

R

, be ause if

f

is su h a fun tion, then for any

t > 0

,

S

t

f

may depend onall oordinates.

Denition 3.1 Let

f

be a fun tion from

S

into

R

, and

R

be a nite subset of

S

. The fun tion

f

is saidtobe mainlylo atedon

R

ifthere existstwo onstants

α

and

β > ρ

su h

that

α > 0

,

β > ρ

and for all

x

∈ R

:

∆

f

(x)

≤ αe

−βd(x,R)

.

(5)

Sin e

β > ρ

, the sum

P

x

∆

f

(x)

is nite. Therefore a fun tion mainlylo atedon a nite set is ne essarilysmooth.

The system weare onsideringwillbesupposed tohave niterangeintera tionsinthe followingsense ( f. Denition4.17, p. 39of [?℄).

Denition 3.2 A parti le system dened by the rates

c

T

(η,

·)

is said to have nite range intera tions if there exists

k > 0

su h that if

d(x, y) > k

:

1.

c

T

= 0

for all

T

ontaining both

x

and

y

,

2.

γ(x, y) = 0

.

The rst ondition imposes that two oordinates annot simultaneously hange if their distan e islargerthan

k

. These ondonesaysthat theinuen e ofasite onthetransition rates of anothersite annotbefelt beyond distan e

k

.

Under these onditions, we provethe following ovarian e inequality.

Proposition 3.3 Assume(2)and(3). Assumemoreoverthatthepro essisofniterange. Let

R

1

and

R

2

be two nite subsets of

S

. Let

β

be a onstant su h that

β > ρ

. Let

f

and

g

be two fun tions mainly lo ated on

R

1

and

R

2

, in the sense that there exist positive onstants

κ

f

, κ

g

su h that,

∆

f

(x)

≤ κ

f

e

−βd(x,R

1 )

and

∆

g

(x)

≤ κ

g

e

−βd(x,R

2 )

_.

Then for all positive reals

s, t

,

sup

η∈X

Cov

η

(f (η

s

), g(η

t

))

≤ C κ

f

κ

g

(

|R

1 | ∧ |R

2 |)e

D(t+s)

_e

−(β−ρ)d(R

1 ,R

2 )

_,

(6) where

D = 2Me

(β+ρ)k

and

C =

2Be

βk

D

1 +

e

ρk

1 − e

−β+ρ

.

(9)

Remark. Shashkin [?℄ obtainsa similar inequality for randomelds indexed by

Z

d

.

We now onsider a transitive graph, su h that the group of automorphisma ts transi-tivelyon

S

(see hapter3 of [?℄). Namelywe need that

•

for any

x

and

y

in

S

there exists

a

in

Aut(S)

, su hthat

a(x) = y

.

•

forany

x

and

y

in

S

andanyradius

n

,thereexists

a

in

Aut(S)

,su hthat

a(B(x, n)) =

B(y, n)

.

Any element

a

ofthe automorphismgroup a ts on ongurations, fun tionsand measures on

X

asfollows:

•

ongurations:

a

· η(x) = η(a

−1

_(x))

,

•

fun tions:

a

· f(η) = f(a · η)

,

•

measures:

R f d(a · µ) = R (a · f) dµ

.

A probability measure

µ

on

X

is invariant through the group a tion if

a

· µ = µ

for any automorphism

a

,and wewantthistoholdforthe probabilitydistributionof

η

t

atalltimes

t

. It will be the ase if the transition rates are also invariant through the group a tion. In order toavoid onfusionswith invarian e inthe sense ofthe semigroup (Denition1.7, p. 10 of [?℄), invarian e through the a tion of the automorphism group of the graph will be systemati allyreferred toas group invarian e inthe sequel.

Denition 3.4 Let

G

be the automorphism group of the graph. The transition rates

c

T

(η,

·)

are said to be groupinvariantif forany

a

∈ G

,

c

a(T )

(a

· η, a · ζ) = c

T

(η, ζ).

Thisdenition extendsinanobviouswaythat oftranslation invarian eon

Z

d

-latti es([?℄, p. 36).

Remark. Observe that for rates whi h are both nite range and group invariant, the hypotheses (2) and (3) are trivially satised. In that ase, it is easy to he k that the semi-group

{S

t

, t

≥ 0}

ommutes with the automorphism group. Thus if

µ

is a group invariantmeasure,thensois

µS

t

forany

t

(see[?℄,p.38). Inotherterms,ifthedistribution of

η

0

isgroup invariant, then thatof

η

t

will remaingroup invariantat alltimes.

4 Fun tional CLT

Ourfun tional entrallimittheoremrequiresthatall oordinatesoftheintera tingparti le system

{η

t

, t

≥ 0}

are identi allydistributed.

(10)

Let

(B

n

)

n≥1

bean in reasing sequen e of nite subsets of

S

su hthat

S =

∞

[

n=1

B

n

,

lim

n→+∞

|∂B

n

|

|B

n

|

= 0 ,

(7)

re all that

| · |

denotes the ardinality and

∂B

n

=

{x ∈ B

n

,

∃ y 6∈ B

n

, d(x, y) = 1

}

. Theorem 4.1 Let

µ = δ

η

be a Dira measure where

η

∈ X

fullls

η(x) = η(y)

for any

x, y

_{∈ S}

. Suppose thatthe transition rates are group invariant. Suppose moreoverthat the pro ess is of nite range, monotone and fullling the requirements of Corollary 2.4. Let

(B

n

)

n≥1

be an in reasing sequen e of nite subsets of

S

fullling (7). Then the sequen e of pro esses

(

N

B

n

t

− E

µ

N

t

B

n

p|B

n

|

, t

_{≥ 0}

)

,

for

n = 1, 2, . . .

onverges in

D([0, T ])

as

n

tends to innity, to a entered Gaussian, ve tor valued pro ess

(B(t, w))

t≥ 0, w∈ W

with ovarian e fun tion

Γ

dened, for

w, w

′

_{∈ W}

, by

Γ

µ

(s, t)(w, w

′

) =

X

x∈ S

Cov

µ

(I

w

(η

s

(x)), I

w

′

(η

_t

(x))) .

Remark. One may wonder wether su h results an extend under more general initial distributions. The point is that the ovarian e inequality do not extend simply by inte-gration with respe t to deterministi ongurations. We are thankful to Pr. Penrose for stressing our attention on this important restri tion. Monotoni ity allows to get ride of this restri tion.

5 Examples of graphs

Besides the lassi al latti e graphs in

Z

d

and their groups of translations, whi h are on-sideredby mostauthors (see [?,?, ?℄),our setting appliestoa broadrange of graphs. We proposesomesimpleexamplesofautomorphismsontrees,whi hgiverise toalargevariety of non lassi al situations.

The simplest example orresponds to regular trees dened as follows. Consider the non- ommutativefree group

S

with nite generator set

G

. Impose that ea h generator

g

is its own inverse (

g

2 _{= 1}

). Now onsider

S

as a graph, su h that

x

and

y

are onne ted if and onlyif there exists

g

∈ G

su h that

x = yg

. Notethat

S

is a regulartree of degree equal tothe ardinality

r

of

G

. The sizeof spheres isexponential:

|{y , d(x, y) = n}| = r

n

. Now onsider thegroupa tionof

S

onitself:

x

· y = xy

: this a tionistransitiveon

S

(take

a = yx

).

From this basi exampleitis possibletoget alarge lass ofgraphs by addingrelations between generators; for example take the tree of degree

4

, denote by

a

,

b

,

c

, and

d

the

(11)

generators, and add the relation

ab = c

. Then, the orresponding graph is a regular tree of degree

4

were nodes are repla ed by tetrahedrons. The spheres do not grow atrate

4 n

:

|{y , d(x, y) = n}| = 4 · 3

n/2

if

n

is even and

|{y , d(x, y) = n}| = 6 · 3

(n−1)/2

if

n

isodd.

b

• db

• da

• d

1 a

b

• _c

dc

• dcd

• _dcdc

•

• dcda

• dcdb

cd

cdc

cdb

cda

• bd

bda

bdb

bdc

Figure 1: Graph stru ture of the tree with tetrahedron ells. The graph onsists in a regular tree of degree 4 (bold lines), where nodes have been repla ed by tetrahedrons. Automorphisms in this graph orrespond to omposition of automorphisms ex hanging ouples of bran hes of the tree (a tion of generator

a

for example) and displa ements in the subja ent regular tree.

6 CLT for hitting times

In this se tion we onsider the ase where

W

is ordered, the pro ess is monotone and satisestheassumptionsinTheorem4.1,theinitial onditionisxedand

f

isanin reasing fun tionfrom

W

to

R

. Inthereliabilityinterpretation,

f (w)

measuresalevelofdegradation fora omponentinstate

w

. Thetotaldegradationofthesysteminstate

η

willbemeasured by the sum

P

x∈B

n

f (η(x))

. Sowe shall fo us onthe pro ess

D

(n)

₌

_{D

(n)

t

, t

≥ 0}

, where

D

(n)

t

= D

B

t

n

is the total degradationof the system attime

t

on the set

R = B

n

:

D

(n)

_t

=

X

x ∈ B

n

f (η

t

(x)).

Itisnaturalto onsidertheinstantsatwhi h

D

(n)

t

rea hesapres ribedlevelofdegradation.

Let

k = (k(n))

be a sequen e of real numbers. Our main obje t is the failure time

T

n

,

dened as:

(12)

In the parti ular ase where

W =

{

working

,

failed

}

(binary omponents), and

f

is the indi atorofa failed omponent, then

D

(n)

t

simply ounts the numberof failed omponents attime

t

,and our system is aso- alled

k

-out-of-

n

system [?℄.

Let

w

0

be a parti ular state (in the reliability

w

0

ould be the perfe t state of an undergrade omponent). Let

η

bethe onstant ongurationwhere all omponents are in the perfe t state

w

0

, for all

x

∈ S

. Our pro ess starts from that onguration

η

, whi h is obviously group invariant. We shall denote by

m(t)

(respe tively,

v(t)

) the expe tation (resp., the varian e)of the degradationat time

t

for one omponent.

m(t) = E[f (η

t

(x))

| η

0 = η] ,

v(t) = lim

n→∞

Var D

_t

(n)

|B

n

|

.

These expressions do not depend on

x

∈ S

, due togroup invarian e. The average degradation

D

(n)

t

/

|B

n

|

onverges in probability to its expe tation

m(t)

. We shall assume that

m(t)

is stri tly in reasing on the interval

[0, τ ]

, with

0 < τ

≤ +∞

(the degradation starting from the perfe t state in reases on average). Mathemati ally, one an assume that the states are ranked in in reasing order, the perfe t state being the lowest. This yields a partial order on ongurations. If the rates are su h that the intera ting parti le system is monotone (see [?℄), then the average degradation in reases. In the reliability interpretation, assuming monotoni ity is quite natural: it amounts to sayingthat the rateatwhi hagiven omponentjumps toamoredegraded state ishigher if itssurroundings are more degraded.

We onsider a mean degradation level

α

, su h that

m(0) < α < m(τ )

. Assume the threshold

k(n)

issu h that:

k(n) = α

|B

n

| + o(p|B

n

|).

Theorem4.1shows thatthe degradationpro ess

D

(n)

shouldremainatdistan e

O(

p|B

n

|)

from the deterministi fun tion

|B

n

| m

. Therefore it is natural to expe t that

T

n

is at distan e

O(1/

p|B

n

|)

fromthe instant

t

α

atwhi h

m(t)

rosses

α

:

t

α

= inf

{t, m(t) = α}.

Theorem 6.1 Under the above hypotheses,

p|B

n

| (T

n

− t

α

)

L

−−−−→

n→+∞

N (0, σ

2 α

),

with:

σ

_α

2 =

v(t

α

)

(m

′

_(t

α

))

2 .

(13)

As in se tion 4, we onsider a transitive graph

G = (S, E)

, where

S

is the set of verti es and

E

⊂

n

{x, y}, x, y ∈ S, x 6= y

o

the set of edges. Fora transitive graph, the degree

r

of ea h vertex is onstant ( f. Lemma1.3.1 in Godsiland Royle[?℄).

For any

x

in

S

and for any positiveinteger

n

, we denoteby

B(x, n)

the open ballof

S

entered at

x

, with radius

n

:

B(x, n) =

_{{y ∈ S, d(x, y) < n}.}

The ardinality of the ball

B(x, n)

is onstant in

x

and bounded asfollows.

sup

x∈S

|B(x, n)| ≤ 2r

n

_{= 2e}

nρ

_{=: κ}

n

,

(8)

where

ρ = ln(max(r, 4)

− 1)

: ompare with formula(4).

Let

Y = (Y

x

)

x∈S

be a real valued random eld. We willmeasure ovarian es between

oordinates of

Y

on two distant sets

R

1

and

R

2

through Lips hitz fun tions (see [?℄). A Lips hitz fun tion is a real valued fun tions

f

dened on

R

n

for some positive integer

n

, for whi h

Lip f := sup

x6=y

|f(x) − f(y)|

P

n

i=1

|x

i

− y

i

|

<

∞.

We will assume the the random eld

Y

satises the following ovarian e inequality: for any positive real

δ

, for any disjoint nite subsets

R

1

and

R

2

of

S

and for any Lips hitz fun tions

f

and

g

dened respe tively on

R

|R

1 |

and

R

|R

2 |

, there exists a positive onstant

C

δ

(not depending on

f g

,

R

1

and

R

2

) su hthat

|Cov (f(Y

x

, x

∈ R

1 ), g(Y

x

, x

∈ R

2 )

| ≤ C

δ

Lip f Lip g (

|R

1 | ∧ |R

2 |) exp (−δd(R

1 , R

2 )) .

(9)

For any nite subset

R

of

S

, let

Z(R) =

P

x∈R

Y

x

. Let

(B

n

)

n∈N

S

su h that

|B

n

|

goesto innity with

n

. Our purpose inthis se tionisto establisha entrallimittheorem for

Z(B

n

)

, suitablynormalized. Wesuppose that

(Y

x

)

x∈S

is a weakly dependent random eld a ording to the ovarian e inequality (9).

In Proposition 7.1 below we prove that, as in the independent setting, a entral limit theorem holdsas soonas

Var Z(B

n

)

behaves, as

n

goestoinnity, like

|B

n

|

( f. Condition (11) below). So the purpose of Proposition 7.2is tostudy the behaviorof

Var Z(B

n

)

. We prove that the limit (11) holds under two additional onditions. The rst one supposes that the ardinality of

∂B

n

is asymptoti ally negligible ompared to

|B

n

|

( f. Condition (7)inse tion4);these ond onditionsupposesaninvarian ebythe automorphismsofthe group

G

,of the joint distribution

(Y

x

, Y

y

)

for any two verti es

x

and

y

. More pre isely we need tohave Condition (10) below,

(14)

a

of

G

.

In ordertoproveProposition 7.1,we shall use someestimations ofBolthausen [?℄that yield a entral limittheorem for stationaryrandom elds on

Z

d

under mixing onditions. Re all that the mixing oe ients used there are dened as follows, noting by

A

R

the

σ

-algebragenerated by

(Y

x

, x

∈ R)

,

α

k,l

(n) = sup

{|P(A

1 ∩ A

2 )

− P(A

1 )P(A

2 )

|, A

i

∈ A

R

i

,

|R

1 | ≤ k, |R

2 | ≤ l, d(R

1 , R

2 )

≥ n},

for

n

∈ N

and

k, l

∈ N ∪ ∞

,

ρ(n) = sup

_{|Cov(Z

1 , Z

2 )

|, Z

i

∈ L

2 (

A

{ρ

i

}

),

kZ

i

k

2 ≤ 1, d(ρ

1 , ρ

2 )

≥ n}.

Under suitable de ay of

(α

k,l

(n))

n

or of

(ρ(n))

n

, Bolthausen [?℄ proved a entral limit theorem for stationary random elds on

Z

d

, using an idea of Stein. In our ase, instead of using those mixing oe ients, we des ribe the dependen e stru ture of the random elds

(Y

x

)

x∈S

in terms of the gap between two Lips hitz transformations of two disjoint blo ks (the ovarian einequality(9) above). Those manners of des ribingthe dependen e of random elds are quite dierent. As one may expe t, the te hniques of proof will be dierentas well(see se tion8).

Proposition 7.1 Let

G = (S, E)

be a transitive graph. Let

(B

n

)

n∈N

S

su h that

|B

n

|

goes to innity with

n

. Let

(Y

x

)

x∈S

be a real valued random eld, satisfying (9). Suppose that, for any

x

∈ S

,

E

Y

x

= 0

and

sup

_x∈S

_kY

x

k

∞

<

∞

. If, there exists a nite real number

σ

2

su hthat

lim

n→∞

Var Z(B

n

)

|B

n

|

= σ

2 ,

(11)

then the quantity

Z(B

n

)

p|B

n

|

onverges in distribution to a entered normal law with varian e

σ

2

.

Proposition 7.2 Let

G = (S, E)

be a transitive graph. Let

(Y

x

)

x∈S

be a entered real valued random eld, with nite varian e. Suppose that the onditions (9) and (10) are satised. Let

(B

n

)

n

be a sequen e of nite and in reasing sets of

S

fullling (7). Then

X

z∈S

|Cov(Y

0 , Y

z

)

| < ∞

and

lim

n→∞

1 |B

n

|

Var Z(B

n

) =

X

z∈S

Cov(Y

0 , Y

z

).

8 Proofs 8.1 Proof of Proposition 3.3

Let

Γ

denote the matrix

(γ(x, y))

x,y∈S

, and let it operate on the right on the spa e of summable series

ℓ

1 (S)

indexed by the denumerable set

S

:

(15)

Γu(y) =

X

x∈S

u(x) γ(x, y).

(We have followed Liggett's [?℄ hoi e of denoting by

Γu

the produ t of

u

by

Γ

on the right.) Thanks to hypothesis (3), this denes a bounded operator of

ℓ

1 (S)

, with norm

M

. Thus for all

t

≥ 0

, the exponential of

tΓ

, is well dened, and gives another bounded operator of

ℓ

1 (S)

:

exp(tΓ)u =

∞

X

n=0

t

n

_Γ

n

_u

n!

.

If

f

isasmoothfun tion,then

∆

f

= (∆

f

(x))

x∈S

, isanelementof

ℓ

1 (S)

. Applying

exp(tΓ)

to

∆f

provides a ontrol on

S

t

f

as shows the following proposition ( f. Theorem 3.9 of [?℄).

Proposition 8.1 Assume (2) and (3). Let

f

be a smooth fun tion. Then,

∆

S

t

f

≤ exp(tΓ)∆

f

.

(12)

It follows immediatelythat if

f

is asmooth fun tion then

S

t

f

isalso smooth and:

|||S

t

f

||| ≤ e

tM

|||f||| ,

be ause the norm of

exp(tΓ)

operatingon

ℓ

1 (S)

is

e

tM

.

A similar bound for ovarian es will be our starting point ( f. Proposition 4.4, p. 34 of [?℄).

Proposition 8.2 Assume (2)and (3). Thenfor any smoothfun tions

f

and

g

andfor all

t

_{≥ 0}

, one has,

kS

t

f g

− (S

t

f )(S

t

g)

k ≤

X

y,z∈S

X

T ∋y,z

c

T

!

Z

t

0 (exp(τ Γ)∆

f

)(y)(exp(τ Γ)∆

g

)(z) dτ.

(13)

In terms of the pro ess

{η

t

, t

≥ 0}

, the left memberof (13) is the uniform bound for the ovarian e between

f (η

t

)

and

g(η

t

)

.

kS

t

f g

− (S

t

f )(S

t

g)

k = sup

η∈X

E

η

[f (η

t

)g(η

t

)]

− E

η

[f (η

t

)]E

η

[g(η

t

)]

.

A slight modi ation of (13) gives a bound on the ovarian e of

f (η

s

)

with

g(η

t

)

, for

0 _{≤ s ≤ t}

. From now on, we shall denote by

Cov

η

ovarian es relativeto the distribution

of

{η

t

, t

≥ 0}

, startingat

η

0 = η

:

(16)

Corollary 8.3 Assume (2)and (3). Let

f

and

g

be two smooth fun tions. Thenfor all

s

and

t

su h that

0 ≤ s ≤ t

,

sup

η∈X

Cov

η

(f (η

s

), g(η

t

))

≤

X

y,z∈S

X

T ∋y,z

c

T

!

Z

s

0 (exp(τ Γ)∆

f

)(y)(exp(τ Γ)∆

S

t−s

g

)(z) dτ.

(14) Proof of Corollary 8.3 . We have, using the semigroup property,

E

_η

_{[f (η}

_s

_)g(η

_t

_{)] = E}

_η

_{[f (η}

_s

_)E[g(η

_t

₎

_{| η}

_s

_{]] = E}

_η

_{[f (η}

_s

_)S

_t−s

_g(η

_s

_{)] = S}

_s

_{(f S}

_t−s

_g)(η).

Also,

E

_η

_[g(η

_t

_{)] = S}

_t

_{g(η) = S}

_s

_(S

_t−s

_g)(η).

Applying (13) at time

s

to

f

and

S

t−s

g

,yields the result.

2

Inordertoapply(14)tofun tionsmainlylo atedonnitesets,weshallneedto ontrol the ee t of

exp(tΓ)

on a sequen e

(∆

f

(x))

satisfying (5). This will be done through the followingte hni al lemma.

Lemma 8.4 Suppose that the pro essis of nite range. Let

R

be a nite set of sites. Let

u = (u(x))

x∈S

be an element of

ℓ

1 (S)

. If for all

x

∈ S

,

u(x)

≤ αe

−βd(x,R)

, with

α > 0

and

β > ρ

, then for all

y

∈ S

,

|(exp(tΓ)u)(y)| ≤ α exp(2tMe

(β+ρ)k

) e

−βd(y,R)

.

This lemma, together with Proposition 8.1, justies Denition 3.1. Indeed, if

f

is mainly lo ated on

R

, then by (12) and Lemma 8.4,

S

t

f

isalsomainlylo atedon

R

, and the rate of exponentialde ay

β

isthe samefor both fun tions.

Proof of Lemma 8.4. Re all that

Γu(y) =

X

x∈S

u(x)γ(x, y).

Observe that if

γ(x, y) > 0

, thenthe distan e from

x

to

y

must beat most

k

and thusthe distan e from

x

to

R

isat least

d(y, R)

− k

. If

u(x)

≤ αe

−βd(x,R)

then:

Γu(y)

_{≤ 2αe}

ρk

e

−β(d(y,R)−k)

M = 2αe

(β+ρ)k

Me

−βd(y,R)

.

Hen e by indu tion,

Γ

n

u(y)

_{≤ α2}

n

e

(β+ρ)kn

M

n

e

−βd(y,R)

.

The result follows immediately.

2

Togetherwith(14),Lemma8.4willbethe keyingredientintheproofofour ovarian e inequality.

(17)

f

and

g

aresmooth. By(14),the ovarian eof

f (η

s

)

and

g(η

t

)

isboundedby

M(s, t)

with:

M(s, t) =

X

y,z∈ S

X

T ∋ y,z

c

T

!

Z

s

0 (exp(τ Γ)∆

f

)(y)(exp(τ Γ)∆

S

t−s

g

)(z) dτ.

Let usapply Lemma 8.4to

∆

f

and

∆

S

t−s

g

.

(exp(τ Γ)∆

f

)(y)

≤ κ

f

exp(τ Me

(β+ρ)k

)e

−βd(y,R

1 )

= κ

f

e

Dτ

e

−βd(y,R

1 )

.

(15)

The lastbound,together with (12), gives

∆

S

t−s

g

(x)

≤ (exp((t − s)Γ)∆

g

)(x)

≤ κ

g

e

D(t−s)

_e

−βd(x,R

2 )

_.

Therefore :

(exp(τ Γ)∆

S

t−s

g

)(z)

≤ κ

g

e

D(τ +t−s)

_e

−βd(z,R

2 )

_.

(16) Inserting the new bounds (15) and (16)into

M(s, t)

, we obtain

M(s, t)

_≤

X

y,z∈ S

X

T ∋ y,z

c

T

!

κ

f

κ

g

e

−β(d(y,R

1 )+d(z,R

2 ))

Z

s

0 e

D(2τ +t−s)

dτ.

Now if

d(y, z) > k

and

y, z

∈ T

, then

c

T

is null by Denition 3.2. Remember moreover

that by hypothesis (2):

B = sup

u∈S

X

T ∋u

c

T

<

∞.

Therefore :

M(s, t)

_{≤ κ}

f

κ

g

Be

D(s+t)

2D

X

y∈S

X

d(y,z)≤k

e

−β(d(y,R

1 )+d(z,R

2 ))

_.

(17)

In order toevaluate the lastquantity, we haveto distinguish two ases.

•

If

d(R

1 , R

2 )

≤ k

,then

X

y∈S

X

d(y,z)≤k

e

−β(d(y,R

1 )+d(z,R

2 ))

_{≤ 2e}

ρk

X

y∈S

e

−βd(y,R

1 )

≤ 2e

ρk

X

n∈N

X

y∈S

e

−βd(y,R

1 )

_I

d(y,R

1 )=n

≤ 4|R

1 |e

ρk

∞

X

n=0

e

(ρ−β)n

≤

4 |R

1 |e

ρk

1 − e

−(β−ρ)

≤ |R

1 |

4e

(ρ+β)k

1 _{− e}

−(β−ρ)

e

−βd(R

1 ,R

2 )

≤ |R

1 |

4e

(ρ+β)k

1 _{− e}

−(β−ρ)

e

−(β−ρ)d(R

1 ,R

2 )

(18)

•

If

d(R

1 , R

2 ) > k

,then wehave,noting that

d(y, R

1 ) + d(z, R

2 )

≥ d(R

1 , R

2 )

− d(y, z)

and that

d(y, z)

≤ k

,

X

y∈S

X

d(y,z)≤k

e

−β(d(y,R

1 )+d(z,R

2 ))

≤

X

d(y,R

1 )≤d(R

1 ,R

2 )−k

X

d(y,z)≤k

e

−β(d(R

1 ,R

2 )−k)

₊

X

d(y,R

1 )≥d(R

1 ,R

2 )−k

X

d(y,z)≤k

e

−βd(y,R

1 )

≤ 4|R

1 | e

ρ(d(R

1 ,R

2 )−k)

e

ρk

e

−β(d(R

1 ,R

2 )−k)

+ 4

|R

1 |e

ρk

X

n≥d(R

1 ,R

2 )−k

e

(ρ−β)n

≤ 4|R

1 | e

βk

1 +

1

1 _{− e}

−(β−ρ)

e

−(β−ρ)d(R

1 ,R

2 )

_.

By insertingthe latter bound into(17), one obtains,

M(s, t)

_{≤ Cκ}

f

κ

g

|R

1 |e

D(t+s)

e

−(β−ρ)d(R

1 ,R

2 )

,

with :

C =

2B

D

e

βk

1 +

e

ρk

1 _{− e}

−β+ρ

.

2

The ovarian einequality(6)impliesthatthe ovarian ebetweentwofun tionsessentially lo atedontwodistantsets de ays exponentiallywiththe distan e ofthosetwosets, what-evertheinstantsatwhi hitisevaluated. Howevertheupperboundin reasesexponentially fastwith

s

and

t

. Inthe ase wherethe pro ess

{η

t

, t

≥ 0}

onvergesatexponentialspeed to its equilibrium, it is possible to give a bound that in reases only in

t

− s

, thus being uniform in

t

for the ovarian eat agiven instant

t

.

8.2 Proof of Theorem 4.1 8.2.1 Finite dimensional laws

Let

G = (S, E)

be a transitive graph and

Aut(

G)

be the automorphism group of

G

. Let

µ

be a probability measure on

X

invariant through the automorphismgroup a tion. Let

(η

t

)

t≥0

beanintera ting parti lesystem fulllingthe requirementsof Theorem4.1. Re all that

{S

t

, t

≥ 0}

denotes the semigroup and

µS

t

the distribution of

η

t

, if the distribution of

η

0

is

µ

.

Proposition 8.5 Let

(B

n

)

n

S

fullling (7). Let assumptions of Theorem 4.1 hold. Then for any xed positive real numbers

t

1 ≤ t

2 ≤

· · · ≤ t

k

, the random ve tor

1 p|B

n

|

N

B

n

t

1 − E

µ

N

B

n

t

1 , N

B

n

t

2 − E

µ

N

B

n

t

2 , . . . , N

B

n

t

k

− E

µ

N

B

n

t

k

(19)

onverges in distribution, as

n

tends toinnity, to a entered Gaussianve tor with ovari-an e matrix

(Γ

µ

(t

i

, t

j

))

1≤i,j≤k

.

Proof of Proposition 8.5 . We will only study the onvergen e in distribution of the ve tor

1 p|B

n

|

N

B

n

t

1 − E

µ

N

B

n

t

1 , N

B

n

t

2 − E

µ

N

B

n

t

2 ,

the general ase being similar. For

i = 1, 2

, we denote by

α

i

= (α

i

(w))

w∈W

two xed ve tors of

R

|W |

. We have,denoting by

·

the usual s alar produ t,

1 p|B

n

|

2 X

i=1

α

i

· N

t

B

i

n

− E

µ

N

B

n

t

i

=

1 p|B

n

|

X

x∈B

n

2 X

i=1

X

w∈W

α

i

(w)(I

w

(η

t

i

(x))

− P

µ

(η

t

i

(x) = w))

!!

=

1 p|B

n

|

X

x∈B

n

Y

x

,

where

(Y

x

)

x∈S

is the randomeld dened by

Y

x

=

2 X

i=1

X

w∈W

α

i

(w)(I

w

(η

t

i

(x))

− P

µ

(η

t

i

(x) = w))

!

=: F

1 (η

t

1 (x)) + F

2 (η

t

2 (x)).

(18)

The purpose is then to prove a entral limittheorem for the sum

P

x∈B

n

Y

x

. For this, we shall study the natureof the dependen e of

(Y

x

)

x∈S

.

Let

R

1

and

R

2

betwoniteanddisjointssubsetsof

S

. Let

k

1

and

k

2

betworealvalued fun tionsdened respe tivelyon

R

|R

1 |

and

R

|R

2 |

. Let

K

1

,

K

2

betwo realvaluedfun tions, dened respe tively on

W

R

1

and

W

R

2

,by

K

j

(ν, η) = k

j

(F

1 (ν(x)) + F

2 (η(x)), x

∈ R

j

),

j = 1, 2.

Let

L

be the lass of real valued Lips hitz fun tions

f

dened on

R

n

, for some positive integer

n

, for whi h

Lip f := sup

x6=y

|f(x) − f(y)|

P

n

i=1

|x

i

− y

i

|

<

∞.

Weassume that

k

1

and

k

2

belong to

L

. Re all that

Cov

η

(k

1 (Y

x

, x

∈ R

1 ), k

2 (Y

x

, x

∈ R

2 )) = Cov

η

(K

1 (η

t

1 , η

t

2 ), K

2 (η

t

1 , η

t

2 ))

But

|K

1 (η

t

1 , η

t

2 )

− K

1 (η

′

t

1 , η

t

2 )

| ≤ 4Lip k

1 X

w∈W

|α

1 (w)

|

X

x∈R

1 |η

t

1 (x)

− η

′

t

1 (x)

|

(20)

Denote

A

1 (W ) = 4Lip k

1 P

w∈W

|α

1 (w)

|

. Then, the fun tions

η

t

1 −→ (Lip k

1 )A

1 (W )

X

x∈R

1 η

t

1 (x)

± K

1 (η

t

1 , η

t

2 )

are in reasing. Hen e, the fun tions

G

±

₁

: (η

t

1 , η

t

2 )

−→ Lip k

1 X

x∈R

1 (A

1 (W )η

t

1 (x) + A

2 (W )η

t

2 (x))

± K

1 (η

t

1 , η

t

2 )

are in reasing oordinate by oordinate. This alsoholds for,

G

±

₂

: (η

t

1 , η

t

2 )

−→ Lip k

2 X

x∈R

2 (A

1 (W )η

t

1 (x) + A

2 (W )η

t

2 (x))

± K

2 (η

t

1 , η

t

2 ).

UnderassumptionsofTheorem2.3andofitsCorollary2.4,theve tor

(η

t

1 , η

t

2 )

haspositive orrelation sothat

Cov

η

(G

±

1 (η

t

1 , η

t

2 ), G

±

2 (η

t

1 , η

t

2 ))

≥ 0.

This gives

|Cov

η

(k

1 (Y

x

, x

∈ R

1 ), k

2 (Y

x

, x

∈ R

2 ))

|

≤ Lip k

1 Lip k

2 X

x∈R

1 X

y∈R

2 Cov

η

(A

1 (W )η

t

1 (x) + A

2 (W )η

t

2 (x), A

1 (W )η

t

1 (y) + A

2 (W )η

t

2 (y)).

From this bilinearformula,we now apply Proposition3.3and obtain the following ovari-an e inequality: for nite subsets

R

1

and

R

2

of

S

, we have letting

δ = β

− ρ

,

|Cov

η

(K

1 (η

t

1 , η

t

2 ), K

2 (η

t

1 , η

t

2 ))

| ≤ C

δ

Lip k

1 Lip k

2 (

|R

1 | ∧ |R

2 |) exp (−δd(R

1 , R

2 )) ,

where

C

δ

is apositive onstant dependingon

β

and not dependingon

R

1

,

R

2

,

k

1

and

k

2

. We then dedu e from Proposition 7.1 that

1 √

|B

n

|

P

x∈B

n

Y

x

onverges in distribution to a enterednormallawassoonasthe quantity

Var

µ

(

P

x∈B

n

Y

x

)/

|B

n

|

onverges as

n

tendsto innity to a nite number

σ

2

. This varian e onverges if the requirements of Proposition 7.2are satised. Forthis, werst he k the ondition of invarian e(10):

Cov

µ

(Y

x

, Y

y

) = Cov

µ

(Y

a(x)

, Y

a(y)

),

a

of

G

and for

Y

x

as dened by (18). We re all that the initial distributionisaDira distributiononthe onguration

η

. Thenithaspositive orrelations. Wehavesupposedthat

η(x) = η(y)

forall

x, y

∈ S

,hen e

a

·µ = µ

andthegroupinvarian e property of the transition rates proves that

µ = δ

η

fullls (19) below and then (10) will hold. Condition(19) is true thanks tothe following estimationsvalidfor any suitablereal

(21)

valued fun tions

f

and

g

,

E

_µ

_{(f (η}

_t

1 )g(η

t

2 ))

=

Z

dµ(η)S

t

1 (f S

t

2 −t

1 g) (η)

=

Z

dµ(η) a

_{· S}

t

1 (f S

t

2 −t

1 g) (η)

sin e

a

· µ = µ

=

Z

dµ(η)S

t

1 ((a

· f)S

t

2 −t

1 (a

· g)) (η)

sin e

a

· (S

s

f ) = S

s

(a

· f)

= E

µ

((a

· f)(η

t

1 )(a

· g)(η

t

2 )) = E

µ

(f (a

· η

t

1 )g(a

· η

t

2 )).

(19)

Hen e Proposition 7.2appliesand gives

σ

2 =

X

z∈S

Cov

µ

(Y

0 , Y

z

)

=

2 X

i,j=1

X

w,w

′

_∈W

α

i

(w)α

j

(w

′

)

X

z∈S

Cov

µ

(I

w

(η

t

i

(0)), I

w

′

(η

t

i

(z)))

=

2 X

i,j=1

α

t

_i

Γ

µ

(t

i

, t

j

)α

j

,

where

Γ

µ

(t

i

, t

j

)

isthe ovarian ematrix asdened inTheorem 4.1; with this we omplete the proof of Proposition8.5.