strong convergence of kernel estimators for product de,nsities of

Nonparametric S/d/rr/ics. Vol. 8, pp. 65 -96

' al' 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under

Reprints available directly from the publisher Photocopying permitted by license only

license by Gordon and Breach Science Publishers Printed in India

STRONG CONVERGENCEOF KERNEL ESTIMATORS FOR PRODUCT DE,NSITIES OF ABSOLUTELY REGULAR POINT PROCESSES LOTHAR

HEINRICHU

and ECKHARD

LIEBSCHERb

"Freiberg

University of Mining and Technology, Institute of Stochastics, Bernhard-von-Cotta-St. 2, D-09596 Freiberg, Germany; bTechnical University of llmenau, Institute of Mathemat-ics, P.O. Box 327, D-98684 llmenau, Germany (Received l3 December1996; Infinalform

21 January 1997)

In this paper we prove almost sure convergence of kernel-type estimators of secondorder product densitiesfor stationary absolutely regular (6-mixing) point processesin Ro. This type of mixing condition can be verified for various classesof point processes under mild additional assumptions.We also obtain rates of convergencewhich mainly depend on the decay of the mixing coefficient and the choice of the bandwidth. In caseof motion-invariant processesthe behaviour of kernel estimators of the pair correlation function is consideredseparately.The results are applibd to kernel-type renewal density estimators. Keywords: Second-order point process; product density; pair correlation function; kernel-type estimator; absolute regularity; large deviation inequalities; estimation variance; rates of convergence; uniform strong convergence AMS I99I subject clssifcations. Primary: 60G55; Secondary: 62G20

1. INTRODUCTION Point procesesare suitable models for the description of randomly or irregularly distributed point-like objects. Their statistical analysis is mostly based on a single observation of a point pattern in a (sufficiently large, convex, compact) sampling window and consists in estimating the first and secondmoment measures.The crucial point 65

66

L. HEINRICH AND E. LIEBSCHER

of this second-orderanalysisconsistsin assuming strict (or at least second-order) stationarity of the underlying point process. The translation invariance of the distribution (or at least of the first and second moment measure) reduces our statistical problem to the estimationof the intensity ) (: mean number of points per unit area) and the reduced secondmoment measurea).j or its Lebesguedensity function (called product densitl,-p{2);.The statistical analysissimplifies still further when isotropy is added to stationarity. In this case the (2) measurecri;j is completely determined by the reduced secondmoment func t ion K ( r ) : a i .| (b (o ,r))l ^ , w h e re b (y ,r) denotesthe cl osedbal l with radius r ) 0 centred at I € R'/, and p(2)(x)(which only depends on the Euclidean norm llxll) is replacedby the pair correlationfunct i o n g ( r ) : Q ( 2 ) ( . r ) / fô r x e R d w i t h l l r l l : r , s a t i s f y i n gK ( r ) : d a , l t d s ,w h e r ei ü d : u a ( b ( o , l ) ) . /ö g(r) s't For the sake of definitenesswe recall some facts from point process theory; for further details and more about statisticsof point processes the reader is referred to the monographs by Daley and Vere-Jones[3], Ripley [22],Karr [17] and Stoyan, Kendall and Meckepal. Let 8'1 (8$) be the o-field (ring) of (bounded) Borel setsof Rd and u4 denotes the d-dimensional Lebesgue measure. A point process V: D, r t 6*, is d e fi n e dto b e a ra n d o m l o c al l y fi ni te counti ng meas ur e on I R u, E ' ] o v e r c e rta i n h y p o th e t i cal probabi l i ty space [f),8, P]. Throughout in this paper we assume that \P is simple ( i. e. e( V ( { , r } ) ( 1 fo r a l l r e R ' ) : l ) a n d stri ctl y stati onary (i .e. ü(B + r) and V(B) have the same distribution for any x e Rd and f e E $). This means that the random closed (discrete) point set s( ü) : { X i. i> l } d e te rm i n e sü c o mp l e te l y. For any E e E o, l et 8*(E) denote the smallestsub-o-fieldof $ that contains all eventsof t h ef o r m { ü ( E n B ) : k } f o r f e E f a n d k : 1 , 2 . . . We are now in a position to formulate a mixing condition ü : L> 1ö;6,which is based on the following absoluteregularity (or see l3-mixing) coe.fficientbetween any two sub-o-fields I,9cÜ, and R o z a n o v V olk ons k ii l27l:

3 ( I , 9 ) : V ( P r e u - P 1x P u ) 1

: ; S U\p L I l P ( A , . B t )- P ( A , ) p ( r r ) l ^/. :rt : l tj _ 1

KERNEL ESTIMATORS FOR POINT PROCESSES

Here V(.) designatesthe total variation norm and the supremum is taken over all pairs of finite partitions {Ar,. . ., A} and { B r , ' ' ' , B t } o f f ) s u c h t h a t A i e X , i : 1 . . . 1 a n d B i e 9 , j - - 1 . . . J. Define E, t: [-r,r]' andEi : : R'\8,. Condition B estimate

Assume that the point processÜ : D,, r 6x admits the

0 ( 8 , u ( s , ) (. E 8 vi * , )=) ( T ) o - ' u * ( r ) r o r a l ls . /> l , where |vO is a non-increasingfunction - the p-mixing rate of ü for satisfyingliml*-pE(r) :0. Ü is called z-dependent, if |vUü:0 some /s € [0,oo). Condition p can be verified for quite a few classesof point processesunder mild additional assumptions.For example, let Ü(.) :: - Xi) be a homogeneousclusterprocessgeneratedby a L>,(V!t)(.) stationary processO : I,> 16y, of cluster centreswith intensity ) and a sequenceü:' ) , i ) l, of i.i.d. copiesof a typical clusterü. having a Pa.s. finite cluster radius

R ": s u p { l l x ül l .: ( { x } ) > o } In Heinrich t10l (see also Heinrich and Molchanov [ 1] for a generalization) the following estimate is derived:

8o(Ei*:,2+)) 0(8v(t')'ü*(85",))S 0 (üo(Ei+,rq)'

+ x d . 2 d( t*ft . 1 ) a - *r ( + + 1 ) d - ' ) r n i r ( R " > t l 4 ) ' In the special case of a Poisson cluster process, that is, Q is a stationary Poisson processwith intensity ), we have g v ( t ) ( c 1E R ! @,

> _rI 4 ) w i th c 1 : ), d 2 a + r(5 o-t + 13d- ' ).

68


Obviously, if in additon P(R,,< ro) : 1, the Poissonclusterprocess is rz-dependent.Similar estimatesof the ,6-mixing rate are known for some classesof (i) dependentlythinned (Poisson) point processes(including hardand soft-core processesas defined by B. MatÖrn and by D. & HStoyan, respectively,see[25]), (ii) point processesassociatedwith germ-grain models (e.g. positive and negativetangent points), seeHeinrich and Molchanov [11], (iii) point processesgeneratedby a (Poisson-)Voronoi tessellationof R" (..g. vertices,midpoint of edges),seeHeinrich [10]. (iv) Gibbsian point processesunder Dobrushin's uniquenesscondit ions , s eee .g . J e n s e n[1 2 ] a n d H e i n ri c h [9]. Condition B with at least polynomial p-mixing rate is needed for proving asymptotic normality of empirical characteristics in point process statistics provided the sampling window expands unboundedly. Of course,Condition B impliesthe ergodicityof the point process iY which in turn is sufficient to prove strong consistency of the empirical (reduced) moment measures,see Karr [17]. By contrast, ergodicity does not suffice to obtain (neither local nor uniform) consistency of kernel estimators of the corresponding product densities. At the end of the first section we recall the definitions of factorial moment measuresand the product densities of any order. Provided that Eü/Ä(B) < oo for all B € Ef. the kth-order factorial moment measure o(u) is deflned by

, ( t )( B r x . . . x 8 6 ): 5

t.

l a , ( t , ). . . 1 u ^ ( " 0 )

.t1...,r€ 1 s(W)

f o r B l, . . . , 8 k € E d Here the sum I- i. taken over all /c-tuplesof pairwise distinct points irt r . . ., xk € s(Ü) The stationarity assumption yields o(t)(.): ûd(.) and admits the desintegationof t(Ä) for k>2 w.r.t. u4,

o ( t ) 1 Bxr. . . x B ü : ^

| " l l l t ( B t - . r )x . . . x ( B r ; - x ) ) u a ( d x ) ,

J 8,.

KERNEL ESTIMATORS FOR POINT PROCESSES ,

(k)

69

where alij is called the reduced kth-order factorial moment measure o n [m a1t - t ) , 9/ ( k - t ) 1 .If rh e -.u ru r. .tff] i s a b sol utel yconti nuous w.r.t. u( r , t ) d; t hen th e c o rre s p o n d i n g R a d o n -N i kodym densi ty p(ft)(on Rd(ft 1)) is called the (reduced)/eth-orderproduct clensityof ü. Note that for a (0-) mixing point processthe generalizedBlackwell th e o remholds whic h i m p l i e sp (t)(r) --- ) a s l l rl i -' co, see[3], p.488. In analogy to Parzen-Rosenblattprobability density estimatorsin classicali.i.d. statistics,seee.g.[23],kernel product density estimators wereproposedfor "increasingdomain statistics"of point processes by D. R. Brillinger [1] (for d: l) and K. Krickeberg [18] for any d> L Under the basic assumption that the stationary point process is Brillinger-mixing,see [17], l7), E. Jolivet [13] derived asymptotically exact bounds of the bias of a kernel estimator for p(k) and of all moments of the absolute difference between a@ and its kernel estimator. Edge-correctedkernel estimators of p(z)(x) and g(r) on a fixed sampling domain were considered by Fiksel [4], see 126) for a recent discussion.Asymptotic normality of kernel product density estimatorscould be proved for Brillinger-mixingpoint processes,see L141,and for Poisson cluster processesunder minimal moment assumptions,see [6], [7]. The main goal of the present paper is to assessthe quality of the kernel estimators for product densities by proving rates of strong (P-a.s.) consistency under mild moment conditions and Condition p. Rates of strong (uniform) convergence have been already obtained in kernel regression estimation, see Collomb and Härdle l2], and in kernel density estimation in the case of a - resp. p-mixing observations,seeLiebscher[16] resp. Yu [29]. The paper is organized as follows: Section 2 contains the main asymptotic results on kernel estimators for second-orderproduct densities of stationary absolutely regular point processes.This concerns firstly the asymptotic behaviour of mean and variance of the estimator under certain smoothnessconditions and a minimal B mixing rate and, secondly,the speedof pointwise as well as of uniform strong convergenceof under various rates of decay of p,!(r) as /---+oo. Corresponding results for two types of kernel estimators for the pair correlationfunctionsof isotropicpoint processes are givenin Section3. At the end of Section 3, the obtained general results are applied to obtain the rate of convergenceof a renewal density kernel estimator. Section 4 deals with large deviation inequalitiesof the Nagaev-Fuk

70

L. HEINRICH

AND E. LIEBSCHER

type for absolutely regular random fields. Section 5 resp. 6 contains the proofs of the results stated in section 2 resp. 3. Throughout, let c,Ct,c2... denote general constants numbered accordins their appearancein the text.

2. RESULTS Throughout in this paper, let ü : I, , r 6x, be a stationary (secondorder) point processon Rd possessing i se.ond-orderproduct density QQ).ln the following we consider a kernel-type estim ator QQ)for Ä p(z) which is defined on the cubic sampling window W,: f},nld by

o l t ) ( .: ' ) ^ärq_,ä*,

rw,,(xt,-(T)

, x€ Rd, ( 2r )

where k: R d r-- R 1 denotes the kernel function and the sequenceof bandwidthsbn ) 0,n e N, is chosen such that lim b,: n-X

0 and lim n bn : @.

(2.2)

n+X

Next we list a set of conditions which are neededto obtain (rates of) local and uniform strong convergenceof the estimator (2.1) to the true function \ p@(x) under the basic Condition p. Condition K(d,p)

Supposethat, for some 0 < Ä ( N,

k(x):0 for llrll> R, sup lk(*)l< *, r l ( r l) S n

t

k@)dx: t. (2.3)

J\o,nl

a n d ,f o r i t , . . . , i t € {1 , . . . d , } , , 1 : 1 , , ,p - | (provided that p> 2),

(2.4)


7I

Examples The following two kernel functions satisfy Condition K(d,2), seee.g.1201, l23l: . d @+ l ) f ( d l 2 )/ . l l r l l\ . k(x):+|l---,,llr,in.R1(.v):..conus..kernel. )ndr2ptt

\

R

/

d('1 ( t -ry) ro,,o,(.r):Epanechnikov 'k(r) '\--l: kerner 4:?,)ly,l1 n d t :p d R_ \

/

C o n d i t ionM 0) S u p p o s eth a t E V' ([0 . 1 ]' /)( rc f or some 1> 4 the product densitiesp(2),p(3),and p(+)exist. ConditionP(r) f o r s o m e6 ) 0 ,

The product density QQ)is continuousat -r € R 't and,

nQ ( u, r' ) < o c a n d

s up

and

a , r ' e ö ( , r . e) U ö ( x , e )

sup

Q6)(r,rr,.r,* u' ) < oc.

u . r , €ö ( , r . : ) . r r , € , 1

1 2s ) ConditionP(K) The product density QQ)is continuouson a compact s e t K C R d a n d , f o r s o m eE > 0 w i t h K , : U . r ,. , < r ( K * , r ) , sup ir,re , K.U(-K')

nQ)(u, v) < cc and

sup

n$) (u,u,)r,+ rr,)< cc.

(2 6)

, r . r , €K , . r r . € R , 1

Pn o p os r r r oN 2. 1. U n d e r C o n d i ti o nKU . 1 ) v ,eh a ve

: ,l{l .ptr,(.r) I p{2)1ry at any continuitypoint , of QQ).If ConrtitionK(d, p) is satisfieclanclpQ)has boundedpartial derivatiyeso.forder p in b(x, e) (.for somee > 0), tlten r plt ) ( r ) : ^ p (2 )(.t)+ o (b l ,) a s n ------+ x

(2.7)

T u B o n B v 2. 1. Let th e C o n d i ti o n sp , K (d . l ),P(r) and C ondi ri on M(4 + 6) -fo, some 6 >0 be satisfted such that lim,-- n2d(4+ö)lö 7v(n): o- T hen, f or r f o ,

)'o?(.'):

(, b,)-'(^n{z)(x) r ( 1 ) ) a sn - -r,c . I*,,n'(r)ay*

(2.8)

12


Relation (2.8) remainsvalidfor x: o with k(y)(k(y) + k(-y)) insteadof the integrandk'(y). In caseof gv(to):0 for some tst 0, (2.8) holds alreadyfor 6:0. Tuponev 2. 2. Su p p o s eth a t th e C o n d i ti o n sP , M(21),K (d,p) and P(*) are satisf.edfor some 1> 2 and somep € N. Further, assumethat QQ)hot boundedpartial derivativesof orderp in b(x,e)for somee>0. C as e ( i) : I f / v ( r ): as n --+ (n)

O(t u )a s r ---+o c fo r s o meq> 2dl l (1 -2),then,

Olt)(r)- ^p(2)("): o(lntt2n(n b,1-atz * ln"'n n ob;d+ bfl) P-a.s. f o r a n yq > ( 1 d + q ) l t @ + q ) , w h e r e0 : ( q \ d - d - t ) - M ) 1 1 @ + q ) (> 0). C a s e( i i ) : I f 0 * Q ) : O ( e x p { - o t b } )a s t - - -x+f o r s o m ea , b } 0 , t h e n , as n ---+ @1

o ? @ ) - ^ a @ ( r ) : o ( t n t t 2 n( n b , y - a t z 1lno2n 71-@t-d-t)11b-d +bp)

p-a.s.

for any nz > (dt - d + b)lh. C a s e( i i i ) : I f 7 v ( t o ) : 0 f o r s o m et s > 0 , t h e n , ans- - +( n , O ? 6 ) - ^ a @ ( " ): o ( t n l t 2 n ( nb ; - a P + lno3n 71-(dt-d-t)ltbnd + bo") p-a.s.

for any rc3) I 11. The latter relation remains validfor 7:2. By equating bp,with each of the other two terms contributing to the rate of convergencewe are able to determine the optimal bandwidth (up to a multiplicative constant) that leads to the best possiblerates in the Cases(i) (iii) consideredin Theorem 2.2.


13

Conor-ranv 2.1. Let the assumptionsoJ'Theorem 2.2 be satisJied. Then. as n---+6. O l t ) ( r )- ) p ( 2 ) ( r ) : o ( b l , )

P-a.s.,

wherethe optimal bandv,idthsminimizing this rate of convergenceare as Jbllows: b, :

I max { (ln"' n ,-')ü , (ln n n-dyt.n1 (ln n n-u)}r} 1 max{ (1n",n "-*it)*,

in Case (i)

| rnu^{ (ln"'r "-tt'iy-,

in Case (iii).

(ln n n-o)le}

in Case (ii)

The next theorem provides the corresponding rates of convergenceof the uniform error

a,(K):

-.r pt2)1x;1. ?:plpl')(')

For this purpose a further smoothnesscondition on the kernel function is needed. Condition L The kernel function k is Lipschrtz continuous, i.e. there is constant L> 0 such that lk(")

- k ( u ) l S L l l " - u l lf o r z , v € R d .

TnBonav 2.3. Supposethat the Conditionsp, M(21),K(d,p), L, and P(K) are sotisrtedforsome1> 2 and somep> I. Further, assumethat QQ)hot boundedpartial derivativesof orclerp on KuJbr some e>0. C a s e( i ) : I f \ v ( t ) :

O ( t , ) o t t - - - +6 J b r s o m eq > 2 d l l ( 1 - 2 ) , t h e n ,

as n ---+6)

L , ( K ) : O ( l n t l 2 n ( nb ) - d t z * l n x ', n -0 1 @ +6i -d (t+ t)l (a r) + b !,)

p - a.s.

q ) . n ' h e r 0e : @ O a - d - t ) - M ) l f o r a n yq > ( t d + q ) l @ + i @ + (d+ q)(> 0). C a se ( ii) I f \ v ( r ) : O (e x p { -a tu } ) o t / ---+o c fo r somea,b } 0,then,, as n ---+6, dl2 A , ( K ) : o ( l n l 1 2 n ( nb , ) 6-a(r+^,)l@+1) + lnRrnn-@:r-d-|)l@+1) + bo") p_a.s. for any nz > (dt - d + b) lb(d + 1).

74


C as e( iii) I f \ v ( to ):0 fo r

(x) s o m e ts > g ,th e n , as n---->

A,(K) :o(lnll2n (n b,;alz + lnRrnn-@t-d-\l@+1)6-d(t+t)l@+t)+ bil

p_a.s.

Jbr any ft > 1l@ + ?). The latter relation is also valid.for 1:2. In quite analogy to the above Corollary 2.1 we get the following CoRorr-aRy 2.2. Let the assumptionsof Theorem 2.3 be satisfied. Then. es n_-+@.

- lp{z)1x;l : o(bl) p-a.s., suplAt2)(") where the optimal bandwidths minimizing this uniform rate of convergenceare asfollows:

b,:

o)}ri} , (ln n n I max{ 1lnü(d+in ,-0:',,r*--L",t-,-, ,t l))N;h)-r-,,(ln 6a n n n-o)h} \ max{(1nxz(d+t)n nz(d+1) max{(ln n (ln n-Ot\-d-l))4o;",",-,, n n-d)}tr} I

in Case(i) in Case(ii) in Case(iii).

Remark 1 For d:l the assumption " n(2) has bounded partial derivativeof orderp" in Prop. 2.l,Theorem 2.2 and 2.3 can be replaced by " nQ)has a Lipschitz continuous (p - 1)st-orderderivative". Remark 2 The above corollaries reveal that in Theorem 2.2 resp. Theorem 2.3 in bot,h Cases(ii) and (iii) the optimal bandwidth is given by bn: ( ln n n- d )a ;4l e a d i n gto pf ) @) - ^p( 2) (r) : o ((l n n n -d 7 # n 1 re s p . L ,(K ):

p r o v i d etdh a t1 > 2 + i + T

o((l n n n-d1#n1

r e s p1. > d + 2 + i + ' #

vergence r a t e se v e nh o l d i n C a s e( r ) i f q > d + 2 + d ( d + l ) l p large enough.

These conand 7 is

3. KERNEL ESTIMATORS FOR THE PAIR CORRELATION FUNCTION AND THE RENEWAL DENSITY First in this section we relate the results of the previous section to motion-invariant (i.e. stationary and isotropic) point processes

KERNEL ESTIMATORSFOR POINT PROCESSES

75

observedin W,. As mentioned in Sect. 1 in this casestatistical secondorder analysis is based on the estimation of the pair correlation function g(r),r ) 0. Two appropriate (univariate) kernel estimators S,Q) and fi,(r) for \'S?) are the following: g ,(r) :

dwaua(W,)b, r d*l

t

r,.(,,rr(W-#=)",r>o

" r 1. - t 2 €s ( Ü )

(3.1) and

,\ : gn\r)

tw,(xr)r,p(llrr_*,ll_r)arr)0. , _ '-, b, dr,tr^w,)k r,.ku,tilr, - ", llo-' \ (3.2)

where k:Rl r-' Rr is an univariate kernel function satislying K(l,1) and b, obeys (2.2).In [a] and126l the univariate Epanechnikov kernel (compare the examplesin Section 2 for d: l) is recommended. PnopostrtoN 3.1. Let k be a s))mmetrickernel function satisfying K(I , 1). If the one-sidedlimits S? - 0) and g(r + 0) existfor somer t 0, then

lim Eg,(r): lim Es,(r): T trt, + 0) + g(r - 0)) n+x) z

I

. l

1 l

n+6)

If Condition K(1, p) is satisfied and g has a Lipschitz-continuous (p - I)st-order derivativegk-tl on lry - €,r2 * el for 0 < rr --12 I n and some e > 0, then

E e , ? ) : \ 2 g ( r )+ ( 1 + r

- t a - r t 1/ ' \{0 < km (d^t- d + b)lb(l + ?) (I + p).


Ca se(iii) I f \ v ( t o) : 0 fo r s o mets ) 0 ,th e n rrTrlx{1nq'nn-##i,

bn :

(ln nn o);i)

for any rtz> IIQ + ?)(1 + p). The latter relationis also validJbr1:

2.

In particular, v,e have 6 n (rt,rr) : O ((l n n n -d l i t' 1 , in the Cases(ii) and (iii) whenever1 > rate remains also valid in Case (i) tf q > enough.

3 + 1+ 3 + fi. rnts convergence d + 2 + (d+ l) lp and1 is large

Remark 3 In general,even if d:1, the pair correlation function g(r) has a discontinuity(of the first kind or a pole) at r:0. For this reason th e relat ions( 3. 4)an d (3 .5 )o f Pro p . 3 .1 a n d th e T h eorems3.1-3.3 do n o t h o ld f or 11: 0. In the second part of this section we specify the foregoing results to s t a t i o n a r y r e n e w a l p r o c e s s e s .F o r t h i s e n d , l e t € r , € 2 , € : , . . . be independent non-negative random variables on [0,8, P] with distribution functions F(x) : P(€r< x) for i > 2 and P(€t < x) :

frf t

- r(fl)dylm, where m: E€z: ff

(1- r(x))dx< n. The

partial sums Xi : €r * . . . * €;,i > I, define a (stationary) simple point process ü : D,>, 6y, on [0,oo) - a so-called (stationary) renewal process associatedwith ,F, see [3] for details. It is well-known that the intensity of ü equals llm and the Lebesgue density h(x) of the measure rl3]f [0, -) n ( )) which coincides with the product density n(z)(x): g(x)lm,x ) 0, is called (Palm) renewaldensity function and takes the form

h(*): i

.f

k_ t

.r(*):-f (*)+ 6 - t)h(t)ctt, J[o' f

(3.6)

where / denotes the probability density of F and f *k is the k-th convolution power of/ Therefore,accordingto (3.1) and (3.2)

: n,@) ;Eä

t ro,a(xi)k (W.#-.)

is an appropriate kernel estimator for h(x)lm.

(3.7)

78


Tnponev 3. 3 . L e t th e k e rn e l fu n c ti o n k: R l t R l sati sfy the Conditions L and K(I,p) -fo, some p> l, and let ,[f "'-t (1 - .F(x))d* < x for some s> (5p + 2)lp. Further, assume that the density of F has a Lipschitz-continuout - I)st-order derivative \p r r : ( l n n l n ) w a n d 0 1 1 1 , ' - t 2{ a , o n l 0 ,a ] , 0 < a 1 x . T h e n , f o b sup V,@) - h(x)lml : o((ln nfI

rtQn+rt1 P-a.s.

(3.8)

111x{12

as n --+ 6.

4. TAIL PROBABILITIES FOR SUMS OF ABSOLUTELY REGULAR RANDOM FIELDS Let ü be a stationary point process satisfying Condition p. Define W , : : 10, t ) o f or a re a l n u m b e r r ) 1 . F u rth er, l et fr:R dt R l be a sequenceof measurable functions with uniformly bounded support, for l l yl l > t. i . e. , t her e ex is tsa fi x e d n u mb e r s > 0 s u c h th a tf,(y):0 Define

S l ' l:

t-

l r , ( x r ) f * ( * r - " r ) f o rn € N , s) 1 .

- x 1x, 2 € s ( Ü )

Obviously, the kernel estimator O? @) coincides with ,Sln)for f,(y) : (nb,)-ot ((y - *)lb,) and s: ll"ll + Rb,. The following lemma provides an exponential inequality for the sum Sl'r similar to that obtained for sums of absolutely regular random sequences,seeYoshihara l28l and Yu [29]. LBuva 4.1. Let ü be a stationarypoint processsatisfying Condition B E(V([0, 1]')tt I n for some1> 2. Further,let m, n, bepositiveintegers suc ht hat 2m 1n a n d p u t t:n f 2 m. T h e n

P ( l s -r E ')^s,;'l>r) I1 4 . Proof of Proposition 4.2. Let E;_f (s) :Uue /_\{_-}E.l(s). Then, by the stationarity ü and Condition p, we get

s u g l P u ' ( , 4-) ( P r * ( u l - ( ,x) )P 8 * ( u 1 1 , ; ; ) ( 1 ) l

AeA)

< Pe(r- 2s) { 0(8v(E,p+,),\v(E5,rz-,)


f o r r ) 6 s , s > l l 4 a n d a l l z € 1 n , ,! e I z , w h e r e ! I " t ' - $ * ( f f ( r ) ) I ü v (fl(r)). Using the properties of the total variation norm V, a successiveapplication of the latter inequality leadsdirectly to (a.3). I Proof of Lemma 4.1. From Prop. 4.2 we obtain that

p ( l z ) - E Z>, ,; )I < p/(t _ t *- ) .- E X ,|;>=,l\I lf \l --.e r,,

|

where xi;,,,2 e l,narei.i.d.,^:'":.:^-rl,

/

^'^::,n.

."*:"0]

distributionas ^S!')for anyy e 12.Thus,we are in a positionto apply Prop. 4.1 to eachof the sumsZ) yielding ( u P(lz) - EZ)i > r) < 2 exp j -c' -----

) !

m d D 2 S t : ))

I

I czr-^'m'e + *d0*Q - 2s). lsi't - E,s,!')lt Finally, remembering that (2*)o: (nlt\d, the inequality

p6sf) - E,s,f')l > ') < t

-. I i

plz,; - EZo;l > "l2o) f l )

c om plet espr oo f o f L e m m a 4 .1 .

tr

5. PROOFS OF THE RESULTS OF SECTION 2 5.1. Proof of Proposition 2.1 By definition of the secondfactorial moment measureand the second product density p(t) the expectation of the kernel estimator Qlt) ir r..n to be equal to - ^ r )'(") \ , ,:

EOf

:^

)

f

;^Wq |

Jm,i

f

, ( x z- x 1 - x \

(--T;J*, Jo,rw,,(rt)k

k1')aQ)@iyb,)dy

,a\

)n"'

(", - x1)dx2dx1


8I

From Condition K(d, l) and Lebesgue'sconvergencetheorem the first assertionof Prop. 2.1 follows immediately.By Taylor expansionof the integrand in b(x,e) (with pth-order remainder term), see Nadaraya 1201,Chapt. II, and making use of Condition K(d, p) and the differentiability properties of p(2), we get the asserted rate of convergencein (2.7). T

5.2. Proof of Theorem 2.1 For the sake of generality we determine the asymptotic behaviour of the variance of

r. t w , ( x r , - ( T )

1

^(2\r'

Qm\ x ) :

, r t U, y ; W

r€Rr

1 1. . x 2€ s ( V )

where t:tn is a non-decreasingsequence of real numbers with | 1 t,1n. ln this proof we consideronly large n for which Rb, < l. PnoposrrroN 5.1. Under the conditionsof Theorem 2.1 with ntz)Q) > 0 we have n 2 ; ( 2 ) / - \ -u

Ant

\^/

( : ) ' D 'O ? ( " ) ( 1 + o ( 1) ) a sn - - +c o

(55)

where the asymptotic behaviourof O2pf,)(") n given in (2.5). A straightforward, but lengthy calculation (see also Heinrich [7], Jolivet [13]) leads to

o'

(,,,ä*,''r""

"'l) :

"')( s'(x1' xz)tsn(xz'rr)) f*,,sn(x1' * o ( t ) ( d x 1 ,d x 2 ) + [

E , ( x 1 ,x z ) ( g n ( x r, 1 3 J

,./Rr'1

I g , ( x t , x r ) f g n ( x zr.: ) * g , ( x : , . x 2 ) ) x r(3)(dx1, rlx2,ad + [

g,(x1. xt)&n(x:,r+)

J R4r

x ( a ( + )- a ( 2 ) x r Q ) ) ( d t t , d x 2 , c l 4 , d x a )

82


and inserting Taking into account the definition of n(Ä),k:2,3,4, : lr,(*r)k((-t, 11 ,) into the latter formula we find xz) 8,(-rr , lb,) that

o'ot) (") : ),(nb,) to(A,,+ "' -yA,t) with

Anl:

o' (T) w,(xr) l*.,,r

: An2

x,)k l*.,,rw,(rr)r,,(,2+ (T)r(=;a)

An3: A,4:

n,tl(x2)dx1ctx2, n,',(x2)dx1dx2,

n"r(x2,x3)ctx1dx2dx3

Io,,,rw,(xrrr(T)r(t#) + xl)k(?) l*r,,rw,(xt)lr,(rs

,(-;t)

x p ( : )( x 2 , 4 ) d x 1 d x 2 d x 3

An5:

tr) |vv,(x2)k(=;t)"(#) f*.,,:w,(xta x o ( 3 ) ( x r ,x : ) d x 1 d x 2 d x 3

An6:

I*,,,'

w,(xt1 tr) l1v,(x3)k(-;-l)

- (+=)

x o ( 3 ) ( - x rx, : ) d x 1 d x 2 d x 3

An7: ^-'

| *,,r

(x3)k(=-) r, (, t)r 1y,

r (t-#=)

x (a(+)Glrr, dx2,dx3. dro) - aQ)(d*r, dx)st(2)(d*r, dxq)) The asymptotic behaviour of the leading term A,1 is obtained by applying the dominated convergencetheorem:

Ant: (tb,)'

l*,0'(y)

p(')6 * yb,)dy

: (rb,)d(nei{') r(t)) I*,0'1)ay* as n---+oc.

(s.6)

83

KERNEL ESTIMATORSFOR POINT PROCESSES Likewise, for x:

o,

, f

p(t)(yb,)dy A,, : b'1., I ua(w,. (w, + yb"))k(y)k(-y) JRd

: 1tb,)d( n"'ol' JIm dko)k(-y)dy +r(1)) (y)1.In caseof xlo For brevity,let tuf- supr,lyl] x. C ombining the latter with (5.6) - (5.9),we obtain in the particular caset: n th e a s s et ionof T heore m2 .1 ,w h i c h i n tu rn , s i n c en (z)(r) > 0, show sthat 7

) \- .a.-,: ( n t b i ) ' o t O f ) ( t ) ( r+ o ( t ) ) a sn - - - +c c .

' '

" t r ^

/ J

Provided that ü is n-dependent the term 8,,2 disappearsso that the existence of EV11E"1 is sufficient. This completes the proof of Pro p os it ion5. 1 and Th e o re m 2 .1 . I Lpvvn 5 . 1 . L e t ü b e o s t c t t i o n o r yp o i n t p r o c e s s satis.f'yi.ng ,.,1.)-. E ü ( 1 0 1, 1 " ) - '< x ; f o r s o n l e1 > l . T h e n , f o rI < t < n a n d r € R'/,

( / \"'

=la',;' (.t) - roi,;' t t)l S .t t L r ,

r .

. l \rt b,,/

(s.13)

y;ith some constantcs hot depenclingon t, n or x. ProoJ'oJ'Lemma 5.1. Using the inequality

l x r+ . . . + X , , , !1 n i ' - 1 \ ) x , l + . ' . + l x , , , l ' ) ,1 > l . m ) 1 and the stationaritv of ü we find that t

''

,{'. r (o),i(.r)- Eqj"'( r)) (2,,)^=lcf1't'11' i

l

Further, with M - S U p . rr .t ,< R l k ( f ' ) 1 ,

(.')I' r loll)

(ffi,) '=

(

(L\

tv ) .1 0 . 1(ü) ' - 6 .,,)(ö (x * r r l ,^ U r r )

E(ü flO,l ) u-t b (o , R b ,)))' ^'. \r ''b'l)

T

86

L. HEINRICH

AND E. LIEBSCHER

The previous lemma combined with Lemma 4.1 and Theorem 2.1 yields P nopos r r r oN 5 .2 . L e t th e C o n d i ti o n s1 3K , (d, l ),P (* ) , and M(21) for Then some 1> 2 be satisfied.

+ n d t-dg ,v Q- 2 (l l " l l + R b ,)) for any r > 0 and sfficiently large t 1n. Moreover, ,f |vQü:0 t0> 0, then

for some

P|O?(') - rO?@)l2 ') < 2d+1exp{- c6r2, db X} + rtr-t ntd+d 6-td for 1> 2.

5.3. Proof of Theorem 2.2 According to the definition of P - a.s. convergence(or by employing the Borel-Cantelli lemma) the relation

(r) : O(o,) P-a.s. as n -+ @ Olt)(")- Ea.1f)

(5.14)

4,, whenever sequence holdsfor a positive(nu11)

t

(") - rO?6)l > ,o,) ( oo P(lpt')

n)l

for some fixed r > 0. From Proposition 5.1 we have

- rpl')(')|>ro,) P(lpt')(") < 2d+1exp{- c6r' ot (n br)o} * cqr-l ant n-td+d'-d+td6td

p*(r- z(ll"ll+ Rb")) + ndr-d


87

Now, in accordancewith the rate of decay of /vQ) we chooset:t,, for the Cases(i), (ii) and (iii) in the following way:

r,:

( (ru*'lnr (ln ln n)2)tl@+q)case(i) case(ii) \ ((d+2)lnnla)ttb ( r o+ z ( l l - l l+ R b , , ) case(iii).

( 5 r. 5 )

This choice guaranteesrhat!n , , ndr;d0v(r, - z(ll-rll + Rb,,))( rc. It remains to find the sequence d, suchthat, for somer)0,

I

.^p{-r2 o1@tt,)o}(

oc

and

or^' n-^td+ttf

D

n)\

d+^td b;^'o < *.

n> I

It is easilychecked that appropriate sequences are

I^

,ll^, 112 n (n b , ;,t12:rln" n n-(t+((t+tllît,l, b-a

an:

for rc>1lr

C a s e s( i ) a n d ( i i )

t 12n (n b n

d ) - d l 2 + l n " n n - d + ( d + t ) l ^bt forrc>llltCases(iii).

lr" By (2.7),

O ? 6) - ) ( , ( 2 )(t) : o (a , + b !)

P-a .s . a s n --+@ '

Finally, after inserting the sequences/i? in the above formulae we obtain the desiredasymptoticrelationsfor eachof the Cases(i), (ii) and (iii). This completesthe proof of Theorem 2.2. I

5.4. Proof of Theorem 2.3 In analogy to (4.2) define

:tD*';,(,), ol')(') I

X'),(*) :

t.

L

r

( n b ,)o. t 1 . - t 2 € s ( Ü )

-

!

1

,

,

88


The compact set K can be coveredby (d-dimensional)cubesCr,...,Cw having edgesof length h andmidpointS il,. . .,u1,, SUChthat l/ I cth J. P ut ei : Ci+ b (o , R ö ,,)a n d

/

|

\ ( l

\n D,)

t'

lw,_.._,(rr)lc, (r, - ", ) .

r1 .,t1€.i(Ü)

on i € { 1,. . ., N} To easethe notation we do not indicatethe dependence on the lhs. From Condition L. we deducethat ll,, ,,ll

- xi,-(t')lS r supll=llvt;," - < sup|X,,,,(r) a . r ' r( ; ,.re'c,ll b, ll

L \'. [dh -,;

(5.16) Y)).. 't-

bn

By the definition of the reduced second-order factorial moment measureand the continuity of oQ) on K it follows that

EY,;, t

(.,) t r, ("' :?u:u'') ä,,1(w,),l!)

and, in analog yto th e p ro o f o f L e mma 5 .1 u si ng E ü([0, I]' )"

E(Y';.)^' t (#,)'' = ( t

(q/ -

1 0 .1 1 ' \,,.,,fiÄ

(s.17)

( oo,

'.,'o ä.,,)(C, * r r r ) (h). (s.r8)

Without loss of generality we may assume that an ------+ 0 as n ---+oo. ' b r ' T o , r . > Put h: e w h e r e E 0 is chosen such that 2dLt/-d cs e ( e r a n + 2 R ) d< l l 6 w h i c h , b y (5 .1 6 ),i mp l i esthat

fr ,/---2

-

-: 2 . s u p X ) , = (-u x)' )n.: (\ r' ',| )< - l -L' r- f- d + f r Yni .-
,",)

'+r ,l\ \

'ct'Li |

\"?

,

'/ t

l?

/

,

/ r i \ " ^ l - 2 ( 1 , '+l l R b , ) ) +|; : : ? i ' ( r \ )

'\

//l

/

,, < t ! e(l t 14,,@l=xi,@)), ,

tr,,,\

'

- tf

/t-3'2'1

\ l - - r / , , \,

'c/1 i-l

r

lC/;i

|

/

/ l n ( ' ylet , x ; , - i-.x\ ) . ( ,.,#) )+l ) /

\'\'tt'l:C1"'

_ t/ , , \ ' ' (;/ ::? "('-2(llr +Rö")) fo r any r ) 0, wher e th e ra n d o m v a ri a b l e s-i ;;-(r),r € 1,,,y e 12,dre independentand Xj,_(.r) has the same distribution as Xt),(*) for all z € Ir,, ! e Iz and any x e K. The first sum on the rhs can be estimated b y a p ply ingP r op. 4.1 , L e m m a 5 .I a n d T h e o re m2 .1 (as i n the proof of Pro p . 5. 1) s o t hat it s u p p e r b o u n d i s a s fo l l o w s : " N 2 ' 1 + r e x p { r ' 1 1 1 2 a 7 , 1 r b , , ) t"'}1*1 1r/ a ; ( : ) " t '

u - / ^ ' n '( 5 . 2 0 )

B y u s i n g( 5 . 1 7 )(,5 . 1 9a) n d Pro p . 4 .1 w e ma y w ri t e that

*t,,-lr,,,) ' ( : vlrr ( t1,-{.i-) l- #) \.\r\/l:(1,,,

\

=r(r'[d*E,,,"-.#;)

90

L. HEINRICH

= ' (Il

AND E .

LIEBSCHER

l

1 \ _ | l- 1tJa.2de/

Y,;,_| t >

\ | :€1_

ct3

\reKl

|

ro-l "

I

for a (sufficiently large) r > 0 i s g u a r a n t e e d w h e n e v earl n d b l > l n n and

s-(t+4

6 - d ( 1 + 11) n - l

Hd(' -t)

ln-" nfor somen)1.

These inequalities hold for 6-a1t+\l0+d) a, - lfi 12n@ b,)-al2 * lnr nt d?r-r)l6+d)n-@t-d-r)10+a) for s om eE > I l 0 + d ), w h e re th e s e q u e n c etn: / i s gi ven by (5.15). Finally, together with a uniform variant of (2.7) we obtain the rates of convergenceof L,(K) as stated in the Cases (i), (ii) and (iii) of Theorem 2.3.

n

91

KERNEL ESTIMATORSFOR POINT PROCESSES

6. PROOFS OF THE RESULTS OF SECTION 3 6.1. Proof of Proposition 3.1 We argue in a similar way as above to prove Prop.2.1. Since Ü is additionally isotropic we have ^ f x

(6.1)

k(x)g(r )- xb,)dx Eg,(r): ^' I -rlb,, J

: ^' I k4)g(r * xbnl(r * +) Es,(r) r J -r1 h , \

o-' ,*

(6.2)

/

As in proving Lemma I in Liebscher [15] we split the integral yields I? ,1 r , , : [ 0- , 1n, + / f . T h e n , l e tti n g n --+ @

':j'ri'],", 'rr1gE''t'' I dxg(r-0)+ [- oG)dxg(r,-0)) Jo

\J-'"

/

it follows (3.3) whence,sincek(x):k(-x), Inserting the Taylor expansion

s (+r x b n: ')D 9 ! s ( ' t ( r* )f f i r "

r ) (* rt x b , )0,( ?