Mixing Properties of Gibbsian Point Processes and Asymptotic ...

Preprint 92 - 051

Mixing Properties of Gibbsian Point Processes and Asymptotic Normality of Takacs-Fiksel Estimates by Lothar Heinrich

Mitgeteilt von F. Götze

30.7.1992

"Diskrete Strukturen in der Mathematik" Sonderforschungsbereich Universität Bielefeld P O R 1 00 l 3 1 D-4800 BielefeldI West Germany Telefon: (0521) 106-4970 Telex: 932362 unibi Telefax: L06-4743 Electronic mail: [email protected] ISSN: 0936-7926 Typeset in Tff,

Mixing Properties of Gibbsian Point Processes and Asymptotic Normality of Takacs-Fiksel Estirnates Lothar HEIIIRICH [Jniversität Bielefeld / Bergakademie Freiberg

Julv 1992 Summary. We prove asymptotic normality of a class of least-squares-typeestirnates in Gibbsian point processesdefined in terms of paramctrized interaction potcntials (in particular pair potentials). This method of parametric inference developed in Takacs(1986) and extended in Fiksel(1988) is based on a single observation in a rectangular sampling window which is assumed to expand unboundedly. In ordcr to apply corresponding CLT's f 0. Now let us introduce the concept of Gibbsian pp's with respect to some local energy E : Rd x M r+ Ftt u {*} being t8d I DJt-rneasurable and the weight processQn. First, for a;rryp € M and distinct 11, .. r nn € Rd let us abbreviate n-l

(1 . 1 )

4 { f , 6 , , , t i ) n @ t ,t D+ E ( * r ,u * 6 , , ) +. . . * E ( r , ,H* I i:7

,",), f:l

if E(Dl:r6,,, p) : E(D?:r6,*(,), p) for everypermutationzr and ü(D?:r6,,, F) otherwiseand EQ,p) 4 O. The Wtt &lJt-measurable mapping,fr, Ml x M *-+Rl U {*}

:A - oo

is called the cond,itional

energyderived from .8. Further d.efineth.e partitior function

z ( E , B , p )A

t exp{*E\[u,

Jnr

FB")}8n(d.rD,

Rß A { t t e M : 0 ( Z ( 8 , 8 , t t )( * } , and for any

B e ag ,Y e IJt

tu1t,,n4=[ t@, B, u')-t Irexp{ -E(b", ttB")}1v(,/tn+ trn.)et(dnb) i f p e R B

nE I I D

L O otherwise G1ötzl(1980a)has shown that the collection of probability kernels {nß}aerg

is a local

specif,cation w.r.t. the family {Rß}re 16 (seePreston(1976),p.16). Deftnition. A probability measureP on lM,W) is said to be the distribution of a Gibbsian point processw.r.t. th,elocal energy.Eanci the rveightprocessQn - briefly P e g(Qu E) -if

(r.2)

PVI'nn.)(t')- n\j',v)

for P-almost every p e M and every E € !9d,Y € rJ,l|r. 3

A necessary and sufficient condition for P to belong t" 9(Qt, -e) were obtained in Glötzl(19S0b)(seealso Nguyen & Zessin(1979)) by imposing an absolute continuity property on the reducedCampbell measureof P defined on l9d I lJI by f C'r(B x Y) A |

J M

f

16(r)1y (,b - 6,)g(dr)P(d,,h) for

|

B eBd,Y €YJr.

JR,d

Theorem 1. (Glötzl(1980b))W" have P e 9(Q,,,,8) itr Ctp satisfiesthe absolutecontinuity conditio" Cp < ^ x P. Under this condition

(1.3)

#fr.;,,,h)-

-E(*,,D} exp{

for (A x P)-almost every (r,rD e Rd x M On the other hand, by application of the theory of desintegrationof measuresone can derive the so-calledrefined Campbelltheorem) see Nguyen and Zessin(1979), Stoyan et al.(1987):Let V ^- P a stationary pp's with intensity )p and / a non-negativemeasurable function on Rd x M. Then (1.4)

t

JRaxM

- ),p f @,,Dc r(d(,,,b)) , l_, l*f @,r-,rDP;(d,tl:)d,r

where the translation ?, is defined by (T"rh)@) - rh(B * r) for B € ts$ t r e Rd, and Pj denotesthe reducedP alm distri,butionof P defined on lM,!lt] by

P,l(r)- )"'- t t rv(r,,h- 6o)g(dr)P(d,,D for y € ylr. JtvtJ1o,t1a A local energy is said to be generatedby lü-body interaction potentialsVw , N ) 2 , (V, - pair potential , Vs - triplet potential) if (1.5)

E ( * , r / , ): a * g f N>2

m \ _'

t.

VN(r,rrt ,uN-r),r€Rd,

'/ r!t...rrN-t€th-6"

where the sum D. it taken over all (,n/ - 1)-tuples r1 ) )rN-r of pairwise distinct atoms of 1b- 6, provided the rhs of (1.5) converges absolutely when ,h i, replaced by ,bn- and BntRd.Herea€R1U{*}denotesthechernzcaIpotentiaIandp> temperature. The mappings Vw , (Ro)N H t r a n s l a t i o n - i n v a r i a n t ,i . e . V x ( * t ) s o t h a t E ( * , r b ) : E ( O , T " I / , ). From now on we suppose that

Rl U {*} are measurable, symmetric zund ,,"t) - Vx(*t - r j... trN - r) t TttT) ) 2 r . , ,€ R d ,

(A0)

P e](Q,t,.E)isastationaryppwithintensity)>0,wherethePoissonweight processQn ir also stationary with intensity \q - 1 and .E is given bV (1.5).

Combiningthis with (1.3) and (1.4) gives f

\ql

f

,\

f

f

l ,rDpÄ@tl:)dr I f @ . , T , r h ) " - E ( 0 , r , ' b ) a r p ( d r b ) : Äl fp@ JR,d

JM

Jw,A J Tw

and finally

P(d,h): ) l r"0h)e-E(0,'D | *"1DP;(d,,h)

(1 . 6 )

for all non-negative measurablefunctions u : M r+ Ft1. We concludeSection 1 by listing some assumptionsto which we will refer severaltimes in the remaining part of the paper. In the following, let llrll resp. lrl denote the Euclidean resp. maximum norm of r € Rd and

d , ( A , B )g i n f { l a - b l : a € A , , be B }

for A,B e Ed

hr is said to have finite interaction range Ro > 0 if (A1)

VN(rr,

, zru) - 0 whenever ,ä?är llrn* ,ill Z Ro ,

anedh,ard-core r & n g e h > 0 i f

Vw(rt,, ,rr) - oo if

(A2)

,#tr

ll*n- *ill < h.

Stabilitv condition: There exists some constant c1 ) 0 such that ,h@)

(A3)

t#

V N ( * r , , " r ) ) - c t r h @ ) f o r a l l B e 8 8 , r b eM .

t.

N-2

r'rt...,rx€thn

Thereexist some rs ) 0, a non-increasingfunction p:lro,oo) ++ [0,*), cz)0suchthat (A4)

- *rll) lVr(*t,rz)l 3 ,rp(ll*t

f o r a l l1 7 t 1 2 € f o r r € T " a n d z 4 -

(46)

whenever ll"t - ,zll ) ,o ,

lVt(*, rr) - Vz(r , r 2 ) l S r , p ( d ( t o ,l , ) ) l l r t

(Ä5)

cse

t z:d(ls,f,))ro

and a constant

_ *rll

Zd with d(fo, f ,) ) ro , äod

p ( d ( r of ,,) \)\( o o

'' (

i.".

,o- - p \ r ) d r . * ) . 1

[*

J,o

t

\

t

\

Here and throughout the paper, for each z -

(tr,--.

,za) € Va and for a fixed number

"y>0 (1 . 7 )

2. Mixing

fr:

d ( G r , G zd ), >( D , E

z€(Jz

F o r e a c hz e U t U U z , l e t A " € I I T y " b e a f i n i t e u n i o n o f p a i r w i s e d i s j o i n t e v e n t s ot h f eform -kt) {p e Mr" : P(Bt) )p(Bt): kt} with B, € t93 n 1,, kr > 0. Using an equivalent

version of the /-mixing condition (2.I) (see Ibragimov and Linnik(1971)) and the fact that öo,o(d(D,,8)) 1öcr,cr(d(Gr,Gz)),it sufficesto find a uniform upper bound of l P ( " z € r r t A , l x z € t f zA " ) - P ( " z € ( r r A " ) l ( w i t h

(2.1I)

P ( * z e ( r z A ,> ) 0)

In order to rewrite (2.1I) .r correlation of two indicator functions fr, f, on M(X ,Mr,) define

f{p) -

tMry(vr)

for i - r,,2.

*\r,re"(?r", rl},. Bv dQ : fzdPlP(fz) (with P(fr) > 0 ) *" get a probability measureon lM,fi] conditional probabilities are given by

whose

(rr"(p'z' A"))-r r A'(9' )Tr"(prz''d?r") if z € (Jz' Q"(ur2,dsr) : { it z e uf . t fr"(prz,dpr,) Therefore,the expressionin (2.rr) is equal to | [ frdP [ frael . By definition of the Vasserstein-Kantorovichmetric, we have for z e Uz f

r

f

f

- flDlTl,(prz,d,rh)rr"(ur2,dp)lAfarl b": | - ]IJ,fur2, - A")-""p |\ J/A " J IL I - " lf@) JM f f

. -

JM

f

f

JA"JM:"

?,th€Mr,

More precisely,by Q.4) , we get

b,ltJälf(z,h) s i n c eb " : 0 f o r z € u i , , 6 " ( f t ) : 0

+1

for z€(Jz.

f o rz € u f a n d 6 " ( / r )s t f o rz e ( J t ,

Theorem 2 yields

(2.r2)

ll,,or- | hdel= tt ,/ä uyl,h)+ r) I

D ,,"

y€Ut z€Uz

Next we have to find bounds of the interaction coefficientsC* to ensurethe validitiy of (2.5). We will use the approachin Klein(1982)and Jensen(1990) although altering some assumptions(in fact Lemma 3.7 in Klein(1982)seemsto be not valid in the stated form) and therefore obtaining slightly different results. In contrast to l(lein(1982) the involved constantsare calculated explicitly.

10

By (A0) it sufficesto estimate the coefllcientsCo, , z € Vd \ {0}. To shorten notation p u t E Q , r , r b p 1 " )! E l t r " , t h r 2 ) a n d E j t , p o r h p , " 1 . ) a E j q " t g r o r , 1 G o u r , ) . ) . N o w w e e m ploy the same trick as in Jensen(1990)(which is referred there to B. Simon) bV introducing a family of probability measures{n!d), 0 ( 0 < t} or [Mr,,,Wy,] bv

- exp{-teilj',e01h10,"}.) IIJtt)@ü + (1 - il80',rb{"}")l}en,,(dp)lZ(il(po,rho,,r,.p,"1"), where 2@(go,rho,rhp,"1) is a normalizingfunction and Q,1,"d.enotesthe stationary PoissonprocessQa (with )q - 1) restrictedto f,. For a measurable/ 1Mr" r- Fülsatisfyinslf @) - f 1Dl Sr(p,rl.) for p,ü e My" set

p}i - f0') - /(0) andH(p,)- E0',?oüp,,\")- E0',rh{,}.).Then r

f

,dH)- n"(bv}" ,dri)l- | [,, F(p)(r!')@p)-nto) {0,4" ( rari)l |l J |M r z F1,)(n,@orh JMr"

- | S1anL?Jrl -nt',(r1n{"et@)ldel -del " | t-l - l l ,f LlnL'tuin) ^z \-)"2 \"'r lJo ao I As shownin Klein(1982)(Theorem3.1.),the Assumptions(A0) - (A4), (AO) and,(2.2) guaranteethe existenceof a constant ca > 0 (dependingorr c1) Ejtr",rbr)

2 (o - grn)p(1,) for all

)cstj,hrRo) such that

p, ?be M.

A careful examination of D. I(lein's proof leads to the following estimate of ca: cg (

D

c r * c a 2 N (t+ 2 R o ,h)* cz IV (l , h)

p(d(to,f ,))

z:d(Ts,f,))1?o

{ e * c a 2 N ( t * 2 R o , o )+ c z c s I f( l , h ) . Therefore

e x p { - l f , l } < 2 ( i l ( ? o , r h o , r b p , , h , V i * j } .

0 if n > IV(1,h).

H e n c eb y t ( p r o , r / 1 r o>) 1 ,

n t D $ ( , jnnD S g r r 2 N ( t + z n o ,{ny)J ä B e B + 2 ( " o - 1 ) } r ( p r , , r h r o ) and

n t q 6 ä l ) S g r r 2 N ( t - r 2 R o ' r k, )u+ r- 1 ) r ( p r o, r h r ) . 72

Combining these estimates and (2.13) with inequality "B - 1 < BeB and keeping in mind the definition of A(., .) and Cor, we obtain the estimate

Co" I gcs2N(t*2Ro,o)(lt/ä*2)BezB

(2.I4)

for l,l E(rhro,0) - grurh(fo) > (o - lkr

* ca))t/(fr).

This leads,in analogy to (2.13), to the estimate

[s,)k4)sIry|,"|,"''.ä6",)||"(ä6",)|d*t...d,n. n)I

Using the representation (2.15) and the assumptions (A4)' (A5), we get

r,2lyJä}p(d(ro ,t "))pr(tr..,rbro) lä(D 6",)l I Bc2nmax{ i:7

yielding

n 5 ' )( l g n D S g r r m a xr{, 2 l1 J ä } p ( d ( r or ,r ) ) o r ( e r , , r hor)

)e ngr1,7, x rlro, vfl')lü(ro)e""/(ro I where ctr :

-a * /Gt

* co).

By exactly analogous arguments,

[n[')(läl)S grrmax{ r,2l7Jä}p(d(ro,rr))p, (vr,,rl,r) elro, I'fro)r",rü(ro) et (drD, I [ u [ ' ) ( l g lS) e l r o ,I V O t ) 1 r " " ü ( reon)@ r D . J

Applying Schwarz' inequality and the Poissonproperty of Qt, *. can write

l

< en@rD vrrt'lll(ro )e"'d(fo) r o)Qr,(drD s 1far(sz)ea-r, ls0h)l'""ü(f J

w h e r eA - A ( o , i l : 7 d " * p { - 2 a * 2 B Q 1 + c o) } . Combining all these estimatesgives r )\/0,t(,f), l f l o ( p , F { o , y } "g, ) - t I o ( r bF, { o , y } "9, ) | < g c r z p ( d ( f o ,f , ) ) p r( V r , , r b o where cr2:

(2.18)

c 2m a x { l , , Z l l t / E }

!/N

+ A)("Lrr'*a) * el+a). Hence

6r(f) 1 gcnp(d(ro,rr)) JA;G\ 15

and,, Dy b ( 4 ,6' /)) - / y '6 r ( f ) ' oo eS ma. r l s n.ow' Cestirr Let US ]a,te b" . ( ot )b''viously, by : 0 for y # z . )

),D

Lett in ng

M Wl" f : Iu.

I rRl sa r,tisfy fy

l f ( p ) - f 1 b )1l r @,,rD,w€can

t,l,o Y,

write

f

dp' , ) - l f ( h ) Q "(pr r 2, d )lr"(pr J

Zrd'Dl

ltr: , dP)n,0tl : , I IU(@, ) - f (bDh(,h)tr,(

I

II"(prZ,h)

d

t;

. I Il*t{ä@Q") +,hG,D+tln(-,0n,0'rr,adn"0'rt,a,D .) TI,(prZ, h)

where the last inequality follows from (2.4). In exactly the same manner as above (using (2.9) and Schwarz'inequality) the numerator in last line is shown to be less than or equal to

with cr3: T/A\/TGTT"too+a)(1 * e*) +'i1o+il. In view of the definitions of l?(.,.) and b, we obtain

-'e@p) b , I , r r 1 / e , r ( h r )[ F , ( u r 2 , ä ) ) J

(2.19)

: c 1 J3A ^ @ r )( r ( n ) ) - ' . Finally, as in the proof of Theorem 3, we need the estimate (2.5) implying (2.17). But this is the essentialpoint in proving the first part of Theorem 4 ; we refer to the abovementionedliterature, e.g. Jensen( 1990). Thus, applying Theorern2 and using Q.1B) , (2.19) we obtain

_ fl Pf(iq h ) - P ( s ) P ( h ) l lJndP_JhdQl t f

f

r

1b,Dr"16r(f) v I Bcpcls

P(h)

7

Assumption (AO) completesthe proof of Theorem 4. tl 3. Central Lirnit Theorems for Gibbsian Point Processes In this section we formulate and prove two CLT's for Gibbsian pp's which satisfy the assumptionsmade in Theorem 3 and Theorem 4, respectively.For this purpose put I r L { - n r " ' , 0 ," . , n l d

and I(, ! t-:,,|)o *t -,;

for a fixedn . umber a ) 0, z eVd ,,

and define

x" L f(T"v),

s, 4 \

w h e r ef , M x o " R l

flr"9,

z€In

is a measurable mapping. We note that the theorems stated below improve earlier results on asymptotic normality of the standardizedsum (2n*7)-a1z(,S" - ES,,) obtained by Jensen(1gg0)and (for Gibbs processeson Zo) by Nahapetian(1980),(1981) and (1991). In particular we could get rid of additional moment assumptions on Xs and the conditions imposed on the rate of decay of the pair potential (which determinesthe behaviour of Ds7 as lrl be weakened.

oo) could

Theorern 5. Let \P ,\,/P € 9(Qn,E) be a Gibbsian pp satisfying the assumptionsof Theorem 3. Supposein addition that (3.1)

[*

, o - ' ( p ( ' ) ) t / , d " ,( o o

t

f'iDP(.,D( oo

Jro

and

(3.2)

,1 Mxo

Then, for a € Rr and 0 < P < go(o,7) (resp. for p > 0 and a ) ooß,i,

w€ have

(2n * r)-a/z(,9"- ES",)2- A[(0,o21 as rz--+oo ,

(3.3)

w h e r e 0 2 : l i m , , - * o oV a r S n l Q " + 1 ) o

:D"ev.o

Cou(f(v), f (7"ü)) and to(p,,j) , go(o.,1)

are chosen exactly as in Theorem 3.

Theorem

6.

Let {/ ,\,/ P € 9(Qn,,E) be a Gibbsian pp satisfying the assumptions of

Theorem 4. Suppose in addition that

(3.4)

nd [* Jn

,o-r p(r)d.r --+ 0

as

n -+ oo

and

(3.5)

f

I

111,[xo

f'(DQ

ildrD ( oo for some a < t 12 . I7

Then relation (3.3) holds for a € pt and 0 a ) at(0,2)),

a'.d p1(t,,1) are taken from Theorem 4.

*h€r€ o1(g,i

Proof of Theorem 5: We begin by showing that (2.5), (2.16) and (3.1) imply

t

(3.6)

Dl!'( oo

zezd

For notational simplicity set p(r) : p(ro) e (0, oo) for 0 ( r ( ro so that, by (2.16),

(3.i)

Co" 1c1ap(d(lo,f"))

for some constant c14depending on a, g,l Since Cy":

Q,u:

C o r - u a n d p ( . ) i s n o n - i n c r e a s i n g ,i t f o l l o w s f r o m ( 2 . 5 ) t h a t

c o s r c r r n"r' c r r - r ,

t Utt"'tUk-t

k

l ' l - ! 2 7

so that inequality (2.10) applies for g - f oT" (on Myo) and h - f oT"+, (on My,_) yielding

D o r p ( ^ azx {_l y l- + , 0 } ) .

l C o u ( X o , XS " )rl , Q , , f f ' ) t v 19

4

Thus, (A.6) and (2.5) give the absolute convergence of o2 -

Dzevd Cou(Xs,Xr).

In

analogy to the foregoing proof we may assume the boundedness of the function f : MTso r-+

Rt, say ll/11"" above-mentioned CLT by Bolthausen(1932) if we suppose the additional condition

[*

,'o-r p(r)dr ( oo implying

Jro

DtrldDo"(oo.

However, ä careful examination of the proof of Bolthausen's CLT for bounded random variables(see also Theorem 7.2.4 in Nahapetian (1991)) revealsthat the weaker cond.ition (3.4) is actually sufficient to (3.3). We will sketch the essentialpart of this proof. For g ) 1, let

*(q) 4 t

kck-'p(tmax{lqlt'l- 1,0})

k>1

Since,by (2.5) and (3.4), Po D qd-'o(q) --+o

as P --+oo

)

q)p

we find a sequence p - p(n) with (3.9)

nd/z

qd-'*(q)

f

--+ 0

and

n-t/'p(n)

-+ 0

as

r? --+ oo .

qlp(n)

Put sy,n:

D

x"

,

z€f n

o2r: Intrsr,,

,

3r:sn/on

,

|r,n:sr,nlon.

Y€In

lv-"l Ao : ?'s,

Lennard-Jones type potentiaL (r" ) n ) 2):

I d'(drlll.ll)" - ,9r(,3'lll"ll)^ I oo if llrll < ä

vl (0,r)

if A.< llrll < Ao

0 if llrll > Ao,

Exponential potential :

v{ (0,r)

:{r

YJ(o,r,y)

i f l l " l l< ä e x p -{ ü 2 l l r l l } i f h < l l r l < l Ao i f l l r l l2 A o ,

I I

oo if rlr

m i n { l l " l l , l l y l l , l-l *y l l }< h

otherwise

0 i f m a x { l l " ll ll ,y l ll,l * - y l l }) A o, 23

o o i f m i n { l l r l l , l l y l l ,-l ly* l l }< h v { ( 0 ,r , u ) :

r 9 1e x p { - ü 2 a ( 0 , r , y ) }

otherwise

0 i f m a x { l l r llll,g r l l ', - y l l }> A o, where a(0, r,y) is the maximal an gle of the iriangle formed by 0 , r, U (seeFiksel (1988)). We shall first deal with the strong consistency of the sequenc. Sn, rL e N. In contrast to the investigation of asymptotic normality below, strong consistency can be established without

any considerations of uniqueness and mixing conditions which means no restric-

tions to the range of a and B in (1.5).

Theorerrt 7. Assumethat O c R" is cornpactanC \et PB e Q(Qn,,Er),ü € O, satisfy (A0) and the following conditions:

(4.5) U(0,r) a DL, U, "{rb)|"-"o(0,'D- "-8" (0,,/) lpt (d,rD)'r o forallf,r€O,0lr. There exist a bounded function a.,: [0, *) * [0, -) a measurablefunction u : M - [0, oo) such that (4.6)

with lim"-,s ,(r)

- 0 and

l

_ ! ""o{-Eo (0,,/)}- "*p{-E'(0,rh)}S ,(llc - ,ll)r(h) I |

foralltS.r€O.tbeM and

(4.7) Inr ut(rD"(DP$ @nb)( oo for i - 1,. - . ,m . Then the sequence of TF-estimate. d,, , fr e N, defined by (4.1) - (4.4) is strongly consistent, i.e.

(4.8)

P r ( { r b € M : l i m r 9 " ( r D - , t } ): t

f o r a l l r g eO

Proof of Tlteorem 7. The idea of this procf consistsin regarding{U"(ü),d e O,fr e N} as a family of contrast functions as they were defined in Heinrich(1991)for general random closedsets. To begin with we additionally assumethat Pd is ergodic. Using the spatial ergodic theoremsof Nguyen and Zessin(1976)and (1979),we get

u:',)\b) - ^ and

lrun(p)P{,@p)

--l*

u. i(, ,,,h)

,n(p)"-U"

24

( o , e )p t

as ??+ oo

@p)

as

n - - +@

f o r P d - a l m o s te v e r y, h e M , ü , r € O a n d i - 7 , . . . . m . T h u s ,b V ( 1 . 6 ) , (4.9)

J*

[Jn(r,rh): tJ(8,r)

for

P8 - almost every ,lt e M , ü,r € O.

Let0 € O ande ) 0 arbitrary but fixed. Then, inviewof the compactnessofO there exist for any 6 > 0 a finite number L - L(6) of points dr,..., $n eO. 4 O \ b(,r,e) such that U!:, b(Si, 6) coversO,. Obviously, L

L

6)}n B;), {ll'9" @ -,ell> €}q U o, u U ({'i"(.,)e b(si, j:t j:7 w h e r eB j : { U " ( $ i , r D - U , ( ü , ( r b ) , r b < ) q}, j :1,...,.t,

a n d q > } i s g i v e nb y

? : m i n { U ( r 9 , r ) ,r e O , } 1 4 ) where (4.5) and the continuity of U(r9,.) (ru a consequence of (a.6) and (4.7)) guar.antee t h e p o s i t i v i t yo f r y .F u r t h e r , i n t r o d u c e t h e e v e n tCsj : { l U . ( , S i , r b ) - U ( r 9 , 0 ) l 5ry} , j: I , " ' , L . S i n c e U . ( $ . ( r D , r D < U . ( 8 , r b ) , t h e d e f i n i t i o no f r l g i v e s t h a t , f o r j - - 1 , . . . , L ,

Bi e {U,(9t.?D,rD > U.(oi,rD -?} n Cj u C; g {U,(l9"(rb),rh)> U(r9,ü) - 2ry}u C; - U(r9,0)l> ry} > 2n}u {lu"(r9i,rh) e {U"(r9,r/,) Hence

(4.10)

-,rll > s})< pr( {u,(,s,,D rr( U {11,9"0h) > zn}) LJ nlN

n)N

L

+ t P r ( U { l u , ( , s i , r-bu)( r e , ü i )>l r y } ) j:t n )N

L

1rr( U U ||,s"(rD- tslll< 6,u,Qs > rtD i,rD - u,(801,),r1,) ,:1

n)Ä/

Taking into account the definition of Pf -a.s.-convergencewe obtain from (4.9) that the first and the second summand on the rhs of (4.10) tend to zero as I/ -) oo. It remains to verify that the third terrn alsc vanishes as Ä- -+ co"

25

By elementary manipulations using (4.6) and Schwarz's inequality we get

- U , ( 8 , , r D l< a ( l l u i- ' 9 , 1 1 ) { r " t O > t( sr (1l -l , y , l*l 2) a ( l l f i- , e l l ) ) + lU,(üi,rD + 2 ( 1 , ( r b ) U , ( r |9) ,t/ ' \ , w h e r el , ( r l , : D L . ( D , .

v n u i ( T " r l t ) r ( T , r l r ), 'i :

!,... ,ffi.

A straigthforwardapplicationof the spatial individual ergodictheorem (seeNguyenand Zessin(1979)) leadsto

r.(,D--+r(rele it

7t'J

as ??-) oo /p , u;(,D,(,Dp'(d,,D)'

for Pd-almost-every ü e M. This combined with (4.9) means that the sequences

- /(,e)l and Yrv(,h) * I.(,liu,(,e,,b) ;lR,lI"(,1, ;lR,

x*(,b) *

converges to zero (as I/ -

oo) in probability w.r.t.

P'e. It is readily checked that the third

term on the rhs of (4.10) does not exceed

> ql,6)]). P'(1(xru(ü) (,(a)* r:äg,(ll'- "ll))+ 2(Yx(,0)'/' + r1,r;) But this probability tends to zero (l/ *

m) whenever 6 > 0 is chosensufficiently small.

Thus, the validity of (a.8) is proved for an ergodic P8. Let us now consider the case when Pr is non-ergodic. Then, arguing as Glötzl and Rauchenschwandtner(1981), there exists a unique weight measute ?üpt on the set of all ergodic Gibbsian pp't {P:,r

e X} w.r.t. the local errergy.Ed such that

f

P0 -

I Pfupo(d*) Jx

,

s e eG e o r g i i ( 1 9 8 8 ) , , C h aeprt 1 4 .

By the first part of the proof, (4.S) holds for all P: ,,r € X, and thus, by ergodic desintegration, also for Pr. This completesthe proof of Theorem 7.X To simplify the writing, let

(0,rDLeG)

#"-Eo

(o,,b)and e&,t)@,rD 4

4 (,e)e #u.1e) and u:r,') ('.e) u]u) oa*

26

#*u.(o)

#a8,"-E'(o,,b)

,

for k,r - 1,...,s.

Theorem 8. Let the Gibbsianpp {/ ,-, P8 e.Q(Qt,Er), ?9€ O, satisfythe assumptions (A0) , (A1) for all N 2 2 and .Ro- al2, (A2) for some .l/ > 2,,(A3), (2.7) and (2.8). Further, supposethat (4.5) and the following conditions are fulfilled: (4.11) I**""?(DPt(d,rD ( oo,i:!,,... ,rn. There exist measurablefunctions a.r;: [0, *)

*

[0, *)

with limr*s c..,6(r)- g

and u6 1Mxo r- [0,oo) (i : I,2) such that ( 4 . 1 2 ) l e ( r l@ , r D - e G ) ? , , r D lS , t ( l l , t - r l l ) r r ( / ) a n d I x r * o u { t D u t ( r D P t ( d , r l t )( o o , (4.13) leQ',t)@,rD- e(r,')(r,rDl S rr(ll0 -- rll)rr(/) and Iy*ou{ch)uz(rDP$ @,rD( oo , and (4.L4) [m*o "{rD(le(t) Qt,rDl + Ie(k,t) (s,rDl)pt (d,rD( oo . (4.15) The matrix S(,t) : (r*r(d))i,,:, given by

a p--riu* "*,(d) n--"" #:%9: 0$ 2

i"fu)1,r;"5')(,r)

*0f t

is non-singular. Then, for a € Rr and 0 < B < 0z(*,7) (resp, for P > 0 and o ) az(9,i, where *z(0,1) and B2(*,1) coincide with ao(0,1) and Bs@,1) in Theorem 3 for c2 _ cs _ 0,Ao r0:

"12), the sequenceof TF-estimates r9,,ir asymptotically normally distributed, more

precisely (4.16)

(2n1a/209*- ,D Z

//(0,.9-1(?9)E(.s)S.-t(tt)) as n ---'oo ,,

for all inner points t9 e O, vrherethe matrix I(d) : (o*t(d))i,,_, is definedby

o6s(d) - '/ 4 ri,o(2nlos*Yg -' i-roo'-

_ I

0r9*

Z

!.

)rJt

u"(ü) 2

rl*)(,1)rl,)(,e),y;;(d)

i,i:l

with

, ! * ) 1 ,g1 ;t u ,.(, rb1) ""\(**))((r *9,,rr\hb, )).P p r' ( dt, r D'-' '^0 8 0 ( 0 ' t Ä \ Jnr

J*r;@)H:P{'(arD,

s tt1!57 I

t

(d,rD, ft1t,rh)fi(,8,T"rDPr

ZidrM . f{ü,rD L I to, (*)";(T,rb- de)- "Ärh)"*"o(0,,D r€rh

Theorem 9. Let the Gibbsial pp \tr'^, Pü e Q(Qn,Er ), ?9€ O, satisfy the assumptions ( A 0 ) w i t h V w : 0 f o r . , ^ r> 3 , ( A 1 ) f o r t / - 2 a n d " R o- a l 2 , ( A 3 ) , ( A 4 ) - ( A 6 ) w i t h

27

?-0: 0, p(r) - 0 for r ) af 2, 1)

2(a* 1) and (2.9). Further, supposethat condition(a.5) of Theorem 7 and the conditions (4.7I) - (4.1a) of Theorem 8 with Qr1 instead of P0 are fulfilIed. Then the assertion (4.16) of Theorem 8 holds for o € Rr and 0 < P < 9r@,1) (resp. for B > 0 and a ) as(p,,7), wher'€o3(0,1) and B3(o,l) are determinedexactly äs o1(g,l) and B1(r,l)

in Theorem 4 for Ro : "12.

Proof of Theorem B. Assume that the true parameter t9 is an inner point of O , that is to say b(d,6) C O for small enoughö > 0. The conditions(4.12) and (4.13) ensureexistence and continuity of the d.erivatin". [/lk)(,1) und,USk'')(,t). In view of the consistency of r9," w e m a y a s s u m et h a t 8 , e b ( ? 9 , 6w) h i c h i m p l i e sU t r ) ( t 9 " ) - 0 , & - 1 , . . . , s .

au"Qt)_ 2 ( 2 n ) - ^ L , \ , n { , , 9 e g w " _ E " ( " , V ) ( u | o , ) ( v ) _ ( J n , i ( . 9 , v ) ) 0r9*

i:l

z€Vn

forj_

1,''' , s. After applying the mean value thecrem to the functicn" tJLr)09),k 1 ,- . . , s , we obtain

(4.r7)

0 _ tiLr)(,.r) (,tf)) I f{,.1,,,- $)U[*,tt

, k * r,... ,s,

l:1

' r r h e r e8 n : ( i u , * ) i : ,

) r s o r n e0 a r l , ( , r 9 g ) : r 1 , 8 , , + ( 1 _ r t ) t 9€ b ( ? 9 , 6f o

1, I - I r . . . r s .

We next determine the asymptotic behaviour of the vecror (ULr\ (d))i:, dom matrix (uLk'tl Qg!))i.,:, as ??---+oo.

and the ran-

Setting

tfiiLu, ü) g e")-o D ui(r"ü/)e(n)(,1 ,7"ü) for k - r,... , s z€.Vn

arrd

u'*.;') (,t,ü) e izn)-d

t

u{7"ü)e(t'l)(c,",{') for k,l _ 1,...,s

Zt Vn

and using the notation of Theorem 8 we can rewrit. ULk)(tl) und,ULk'')(,1)as follows: 1 ._. -irtn(,e) : (2")-o t D "!lJf,t,ü)fo($,r"v) zQlln

i=.1

and

-*r|*,')(,r) : t(,f;',(,e,v - "f,)(e,v),L'], (,e, ü)) r,("e,r"ü) )(2,)-o L z:l zeVr. .-]o

Since the assumptionsof Theorem B imply uniquenessand ergcdicity of \tri^, PB, we find by means of the ergodic theoremsof Nguyen and Zessin(1976)I G-979)together with (1.6) that

* '5*)('l) f, (0,7,/) * 0 and "*,1@,rh)

(2")-oI z€Vn

f o r P d - a l m o s t e v e r yr h e M

,i:

l-,...,rn and k -\...;s.

Thisandthestrong consis-

tency of the sequence Sntn € N conrbined with the moment and continuity assumptions (4.12) - (4"14) lead to

- Dr!*)1,r;,1')(,r) -. srr(re) -tgl zir;rrr(,tg)) r.-+oo

i:l

for P8-almcst every ,h e M and k, I * 7t... ts. t'herefore, using (4.17), ihe li,miting distributions of the vectors *(U[r>(d))i:r,

and (tz')-d/2 t (fzrl-d/2i(,9,,, - rel lr*,(d)) ' *-' ' *-, |

{$,,r"*));=, t rl*)(,.y).r, r

"€vn i-r

coirrcide To accomplish the proof of Theorern 8 we appl;r Theor em 5 to the furrction

tf 0 h )a- t " *r f

n

' 5 * ) ( , . 1 ) f i ( , t , r' rhy)i t h ( o r , - - .t a s )e R " \ { 0 } ,

i:l

*:1

which, in vier,vof (a.11) and the hard-core property, satisfies (3.2). By the wäy, since the interaction potentials have finite range, Ds, decreasesexponentially fast to zero as lrl - oo so that V ^. p(o) is exponentially /-mixing. With the above notation Theorem 5 vields

(4.18)

(2n)-d/2D t(r"ü) -?- ^r(0,i z€Vn

which by Cram6r-Wold

"* ap7,1(ü))., ,, , oo:

*,1-1

arguments proves that

Thus, together with (4.15) we get (4.16) which completesthe proof of Theorem 8. n Tl:,e assertion of Theorem 9 can be proved by quite similar arguments based on Theorem 6.

29

Acknowledgement. This paper was written during the author's stay (frcm August 1990 till July 1992) at the SFB "Discrete Structures in Math,ematics"of the University of Bielefeld. I would like to thank the DFG (German ResearchFoundation) for supporting this work. Referenees Baddeley,A. and Moller, J. (1939). Nearest-neighbourMarkci'point processesand random sets. Internat. Statist. Rev. 57 , Bg - 121. Bolthausen, E. (1982). On the central limit theorern fcr statronary mixing random fields. Ann. Probab. 10 , 1049 - 1052. Cressie,N. (1991). Statisticsfor Spatial Data. Wiley & Sons , New York. Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan,D. and Tanemura,M. (1992). On parameter estimation for pairwise inte::actionpoint processes.Ann. Inst. Statist. Math., to appear. Dobrushin, R.L. (1968). I)escription of a random field by means of conditional probabilities and conditions of its regu.larity. Theor. Prob. AppI. 13 , 797 - 224. Fiksel, T. (1988). Estimation of interaction pctentials of Gibbsian point processes. statist,ics19 ,'77 - 86. F'öilmer.H. (1982). A covarianceestimate fr:r Gibbs measures"J. Funct. Anal. 46, 387 - 395. tleorgii, H.-O. (1988). Gibhs Nfeasuresand P.haseTransitions. de Gruyter , Berlin. Glötzl, E. (1980a). Bemerkungenzu einer Arbeit von O. I(. Kozlov. Math..lfachr. 94, 277 - 289" Glötzl, E. (1980b). Lckale Energien und Fctentiale für Punktprozesse.Math. I{achr. gG, 195 - 206. Glötzl, E. and Rauchenschwandtner, R.(1981). On the statisticsof Gibbsian processes. In: The First Pannonian Symp. on Math. Statistics(eds. P.Revesz,L.Schmetterer, V.M. Zolotarcv) Lecture Notes in Statistics Vol. 8, Springer, Berlin, pp. 83 - 93. Guyon, X. and Künsch, H.R. (1992). Asymptotic comparisonof estimators in the Ising model. In: Proc. of I.A.C. Conference"stochastic Models, Statistical Methods and Algorithms in Image Analy'sis",Roma, 1990 (eds" P. Barone, A. Frigessi, lvl. Piccioni), To appear in Lecture Notes in Statistics, Springer, Berlin Guyon, X. (1991), MdihoCes de pseudo-vraisemblance et de codagepour le processus

3ß

ponctuelsde Gibbs. Prdpublicationdu SANIOS (Univ. F'aris1) Nq. 6, Nov. 199i. Heinrich, L. (1991). Asymptotic properties of mimimum contrast estimatcrs for parameters of Boolean models. Metrika sr:bmitted, Preprint No" 91-066, SFB "Discrete Structures in Mathematics", University of Bielefeld. Ibragimov, I.A. and Linnik, Yu.V. (1971). Indeperidenta,nd Stationary Sequencesof Random Variable.s.Wolters-Nocrdhoff, Grcningen. Jensen,J.L. (1990). Asymptotic normality of estima.;esin spatial point processes. ResearchRepc,rtsNo. ztD, Dept. of Theoretical Statisiics, Inst. of Mathematics, University of Aarhus. Jensen,J.L. and Moller, J. (1991)" FseudolikelihooCfor exporrentiaifarnily models of spatial point processes. Ann. AppL Probab. L , 445 - 46L Kiein, D. (1982). Dobrushin uniquenesstechniquesand the decay of correlationsin continuum statistical mechanics. Commun. Math. Phys. 86 , 227 - 246" Kirnsch, H. (1932). Decay of correlationsunder Dobrushin's uniquenesscondition and its applications. Ccmmun. L[ath. Ph;6. 84 , 207 - 222. Mase, S. (1992)" IJniform LAN condition of planar Gibbsian point processesand. optirnality of maximum likelihood estimators of soft-corepotential ,functicns. Probab. Theory P.elat. Fields 92, 57 - 67. Nahapet:.an,B.S. (1980). The central limit theorem for random fields with mixing property. In: Multicomponent Random Systems(eds. R.L. Dobrushin, Ya.G" Sinai), Advancesin Probability and Related Topics, Vol. 6, Marcel Dekker, New York - Basel, pp. 531 - 547. Nahapetian,E.S. (1931). Limit theoremsand statistical mechanics.In: Random Fields (eds. J .Füt z, J.I.Lebowitz, D.Szasz),Colloquia Mathematica Societatis Jänos Bolyai 27. , North Holland, Amsterdam, pp. 799 - 820. Nakhapetyan, B.S. (1987). An approach to proving limit theoremsfor depending randcm variables. Theor. Prob" Appl. 32, 535 - 539. Nahapetian, B. (1991). Limit Theoremsand Some Applications in Statistical Physics. TEUBNER-Texte zlurMathematik, Band L23, B.G" Teubner Verlagsgesellschaft,Stuttgart - Leipzig. Nguyen, X.X" und Zessin,H. (1976). Punktprozessemit Wechselwirkung.Z. Wahrscheinfichkeitstheorieverw. Gebiete 37 ,91- 126. Nguyen,X.X. und Zessin,H" (1979). Ergodic theorernsfor spatial processes.

31

Z. Wahrsckein.lichkeitstheor;everw. Gebi.ete48 , i33 - 158. Nguyen, X.X. and Zessin,tL (1979). Integral and.d.ifferential. characterizationsof t,he Gibbs proces-q.Math" .ldachr.88 , i05 - 115. Preston, C. (i97"6)" Rantiotn Fielc's. Lect" Notes in lvlathematicsVol. 534, Springer, Berlin. Stoyan, D., Kend.all,W.S. , Mecke, J. (i987). SüochasticGeametry and Lts A,pptr"ic st:ons" Aka,Cesrie-rvrerlag, E erlin. Takacs,fi.. (1986)" Esti:nator for the pair-potential of a Gibb,sianpoint process. Math. Operafionsforscä. u. Statist., ser. statist,. 1.? , 429 - 433.

Fachbereich Mathematik BergakademieFreiberg Bernhard-von-Coi ta- Str. 2 D- C)-9200 Freiberg(Sachsen) Germany

32