Normal Approximation in Geometric Probability

Transcription

1 Normal Approximaion in Geomeric Probabiliy Mahew D. Penrose 1 and J. E. Yukich 2 Universiy of Bah and Lehigh Universiy February 2004 Absrac Saisics arising in geomeric probabiliy can ofen be expressed as sums of sabilizing funcionals, ha is funcionals which saisfy a local dependence srucure. In his noe we show ha sabilizaion leads o nearly opimal raes of convergence in he CLT for saisics such as oal edge lengh and oal number of edges of graphs in compuaional geomery and he oal number of paricles acceped in random sequenial packing models. These raes also apply o he 1-dimensional marginals of he random measures associaed wih hese saisics. 1 Deparmen of Mahemaical Sciences, Universiy of Bah, Bah BA1 7AY, Unied Kingdom: m.d.penrose@bah.ac.uk 2 Deparmen of Mahemaics, Lehigh Universiy, Behlehem PA 18015: joseph.yukich@lehigh.edu 2 Research suppored in par by NSA gran MDA and NSF gran DMS

2 1 Inroducion and main resuls In he sudy of limi heorems for funcionals on Poisson or binomial spaial poin processes, he noion of sabilizaion has recenly proved o be a useful unifying concep [4, 9, 11]. Laws of large numbers and cenral limi heorems can be proved in he general seing of funcionals saisfying an absrac sabilizaion propery whereby he inserion of a poin ino a Poisson process has only a local effec in some sense. These resuls can hen be applied o deduce limi laws for a grea variey of paricular funcionals, including hose concerned wih minimal spanning ree, neares neighbor graph, Voronoi and Delaunay graph, and packing (see Secion 2). Several differen echniques are available for proving general cenral limi heorems for sabilizing funcionals. These include a maringale approach [9] and a mehod of momens [4]. In he presen work, we revisi a hird echnique for proving cenral limi heorems for sabilizing funcionals on Poisson poin processes, which was inroduced by Avram and Bersimas [1]. This mehod is based on he normal approximaion of sums of random variables which are mosly independen of one anoher in a sense made precise via dependency graphs, which in urn is proved via Sein s mehod [12]. I has he advanage of providing he possibiliy of explici error bounds and raes of convergence. We exend he work of Avram and Bersimas in several direcions. Firs, whereas in [1] aenion was resriced o cerain paricular funcionals, here we derive a general resul holding for arbirary funcionals saisfying a sabilizaion condiion which can hen be checked raher easily for many special cases. Second, we consider non-uniform poin process inensiies and do no require he funcionals o be ranslaion invarian. Third, we improve on he raes of convergence in [1] by making use of he recen refinemen by Chen and Shao [6] of previous normal approximaion resuls for sums of mosly independen variables. Finally, we apply he mehods no only o random variables obained by summing some quaniy over Poisson poins, bu o he associaed random poin measures, hereby recovering many of he resuls of Baryshnikov and Yukich [4] on convergence of measures, wih exra informaion abou he rae of convergence and wihou requiring higher order momen calculaions. Le ξ(x; X ) be a measurable R-valued funcion defined for all pairs (x, X ), where X R d is finie and where x X. When x / X, we abbreviae noaion and wrie ξ(x; X ) insead of ξ(x; X {x}). For all > 0 le ξ (x; X ) := ξ(x; x + 1/d ( x + X )) 2

3 where given a > 0 and y R d, we le ax := {ax : x X } and y + X := {y + x : x X }. We say ξ is ranslaion invarian if ξ(x, X ) = ξ(y+x, y+x ) for all y R d. When ξ is ranslaion invarian, he funcional ξ simplifies o ξ (x; X ) = ξ( 1/d x; 1/d X ). Given a probabiliy densiy funcion κ wih compac suppor A R d, for all > 0 we le P := P κ denoe a Poisson poin process wih inensiy κ on A. We shall assume hroughou ha κ is bounded wih supremum denoed κ. The following noion of exponenial sabilizaion, inroduced in [4], plays a cenral role in all ha follows. For x R d and r > 0, le B r (x) denoe he Euclidean ball cenered a x of radius r. Definiion 1.1 ξ is exponenially sabilizing for κ if for all 1 and all x A, here exiss an a.s. finie random variable R := R(x, ) (a radius of sabilizaion for ξ a x) such ha ξ (x; [P B 1/d R(x)] X ) is independen of X for all finie X A \ B 1/d R(x) and here exiss a consan C > 0 such ha for all > 0 sup P [R(x, ) > ] C exp( /C). 1, x A Definiion 1.2 ξ has a momen of order p > 0 if sup E [ ξ (x; P ) p ] <. (1.1) 1, x R d For > 0, define he random weighed poin measure µ ξ := x P ξ (x; P )δ x. and he cenered version µ ξ := µξ E [µξ ]. Le B(A) denoe he se of bounded Borel-measurable funcions on A, and le B c (A) denoe he se of coninuous funcions in B(A). Given f B(A), le f, µ ξ := A fdµξ and f, µξ := A fdµξ. Le Φ denoe he disribuion funcion of he sandard normal. Our main resul is a normal approximaion resul for f, µ ξ, suiably scaled. Theorem 1.1 Le ξ be exponenially sabilizing and assume ha ξ saisfies he momen condiion (1.1) for some p >. Le f B(A) and pu T := f, µ ξ. There exiss a finie consan C depending on d, ξ, κ and f, such ha for all > 1 [ ] sup P T E T (VarT ) Φ() 1/2 C(log )d (VarT ) /2. (1.2) R

4 Remarks (i) For many funcionals of ineres Var f, µ ξ = Θ() (see Remark (v) below). Whenever Var f, µ ξ = Θ(), Theorem 1.1 yields a rae of convergence O((log )d 1/2 ) o he normal disribuion. We are no sure if he logarihmic facors can be removed. The rae in [1] is O((log ) 1+/(2d) 1/4 ). (ii) If in Theorem 1.1 we assume only ha (1.1) holds for some p > 2 insead of some p >, and if VarT = Θ(), hen he proof of Theorem 1.1 can be adaped o give a rae of convergence of O( 1+ε p/2 ), for arbirary ε > 0. (iii) We do no have rae of convergence resuls in he binomial (non-poisson) seing. cenral limi heorems in he binomial seing, we refer o [9] and [4], which rea uniform and non-uniform samples respecively. (iv) Some funcionals, such as hose defined in erms of he minimal spanning ree, saisfy a weaker form of sabilizaion bu are no known o saisfy exponenial sabilizaion. In hese cases univariae and mulivariae cenral limi heorems hold [8, 9] bu our Theorem 1.1 does no apply and explici raes of convergence are no known. (v) In many cases, combining Theorem 1.1 wih known resuls on he asympoic behavior of Var(T ) yields cenral limi heorems. More precisely, i is esablished in Theorem 2.4(i) of [4], using mehods developed in [9, 11], ha if A is convex, κ is coninuous, and ξ lies in a cerain class of slowly varying funcionals SV(4/) which includes all ranslaion invarian funcionals as a special case, and if cerain exponenial sabilizaion and ph momen condiions hold which are similar in spiri o hose given in Definiions 1.1 and 1.2 above, for some p > 2, hen hen for all f B c (A), lim 1 Var f, µ ξ = f(x) 2 V ξ (κ(x))κ(x)dx (1.) A wih V ξ ( ) given explicily in erms of ξ in [4]. Combining (1.) wih Theorem 1.1 yields ( ) f, 1/2 µ ξ D N 0, f(x) 2 V ξ (κ(x))κ(x)dx, A where N (0, σ 2 ) denoes a cenered normal disribuion wih variance σ 2 if σ 2 > 0, and a uni poin mass a 0 if σ 2 = 0. Thus, when (1.) holds we can use Theorem 1.1 o recover he conclusions of Theorem 2.4 (ii) of [4] (a cenral limi heorem for he finie-dimensional disribuions of he random field ( f, 1/2 µ ξ, f B c(a))), wihou any compuaion of higher order momens. This is characerisic of Sein s mehod. For 4

5 (vi) Our Theorem 1.1 requires neiher he underlying densiy funcion κ nor he es funcion f o be coninuous (boh of hese condiions are imposed in [4]). In paricular, Theorem 1.1 applies when f is he indicaor funcion of a Borel subse B of A, giving normal approximaion for µ ξ (B). 2 Applicaions Applicaions of Theorem 1.1 o geomeric probabiliy include funcionals of proximiy graphs, Boolean models, and random sequenial packing models. The following examples are for illusraive purposes only and are no mean o be encyclopedic. For simpliciy we will assume ha R d is equipped wih he usual Euclidean meric. However, since we do no assume ha ξ is ranslaion invarian, he examples can be modified o rea he siuaion where R d has a local meric srucure. 2.1 k-neares neighbor graph Le k be a posiive ineger. Given a locally finie poin se X R d, he k-neares neighbors (undireced) graph on X, denoed kng(x ), is he graph wih verex se X obained by including {x, y} as an edge whenever y is one of he k neares neighbors of x and/or x is one of he k neares neighbors of y. The k-neares neighbors (direced) graph on X, denoed kng (X ), is he graph wih verex se X obained by placing a direced edge beween each poin and is k neares neighbors. Le N k (X ) denoe he oal edge lengh of he (undireced) k-neares neighbors graph on X. Noe ha N k (X ) = x X ξk (x; X ), where ξ k (x; X ) denoes he sum of he edge lenghs in kng(x ) inciden o x. If A is convex or polyhedral and κ is bounded away from 0 on A, hen ξ k is exponenially sabilizing (cf. Lemma 6.1 of [9]) and has momens of all orders. Moreover Var[N k ( 1/d P )] C. We hus obain he following raes in he CLT for he oal edge lengh of N k ( 1/d P ) improving upon Avram and Bersimas [1] and Bickel and Breiman [5]. A similar CLT holds for he oal edge lengh of he k-neares neighbors direced graph. Theorem 2.1 Suppose A is convex or polyhedral and κ is bounded away from 0 on A. Le N := N k ( 1/d P ) denoe he oal edge lengh of he k-neares neighbors graph on 1/d P. There exiss a finie consan C depending on d, ξ k, and κ such ha [ ] sup P N E N (VarN ) Φ() 1/2 C(log )d 1/2. (2.1) R 5

6 Similarly, leing ξ s (x; X ) be one or zero according o wheher he disance beween x and is neares neighbor in X is less han s or no, we can verify ha ξ s is exponenially sabilizing and ha he variance of x 1/d P ξ s (x; 1/d P ) is bounded below by a posiive muliple of. We hus obain raes of convergence of O((log ) d 1/2 ) in he CLT for he empirical disribuion funcion of k neares neighbor disances on 1/d P, improving upon hose implici on p. 88 of [7]. Using he resuls from secion 6.2 of [9], we could likewise obain he same raes of convergence in he CLT for he number of verices of fixed degree in he k neares neighbors graph. 2.2 Voronoi and sphere of influence graphs We will consider he Voronoi graph for d = 2 and he sphere of influence graph for all d (see secions 7 and 8 of [9] for definiions). From he resuls of [4, 9, 11], we know ha he oal edge lengh of he Voronoi and sphere of influence graphs on X boh admi he represenaion x X ξ(x; X ); moreover, if κ is bounded away from 0 and infiniy and A is convex, hen ξ is exponenially sabilizing and saisfies he momen condiion (1.1) for all p > 1. Also, he variance of he oal edge lengh of hese graphs on P is bounded below by a muliple of. We hus obain O((log ) d 1/2 ) raes of convergence in he CLT for he oal edge lengh funcionals of hese graphs on P, hereby improving and generalizing he resuls of Avram and Bersimas [1]. For he sphere of influence graph we may draw on he resuls of secions 7.1 and 7. of [9], o obain O((log ) d 1/2 ) raes of convergence in he CLT for he oal number of edges and he number of verices of fixed degree in he sphere of influence graph on P. 2. Random sequenial packing models The following prooypical random sequenial packing model is of considerable scienific ineres; see [10] for references o he vas lieraure. Wih N() sanding for a Poisson random variable wih parameer, we le B,1, B,2,..., B,N() be a sequence of d-dimensional balls of volume 1 whose ceners are i.i.d. random d-vecors X 1,..., X N() wih probabiliy densiy funcion κ : A [0, ). Wihou loss of generaliy, assume ha he balls are sequenced in he order deermined by marks (ime coordinaes) in [0, 1]. Le he firs ball B,1 be packed, and recursively for i = 2,,..., le he i-h ball B,i be packed iff B,i does no overlap any ball in B,1,..., B,i 1 which has already been packed. If no packed, he i-h ball is discarded. In much of he lieraure, he ime coordinaes are assumed independen of he spaial coordinaes bu since we do no need o confine aenion o ranslaion invarian models, 6

7 we do no require his assumpion here. Packing models of his ype arise in diverse disciplines, including physical, chemical, and biological processes. See [10] for a discussion of he many applicaions, he many references, and he widespread use. Penrose and Yukich [10] esablish he asympoic normaliy of he number of acceped balls when he spaial disribuion is uniform and also show [11] a LLN for he number of acceped balls when he spaial disribuion is non-uniform. For any finie poin se X R d, assume he poins x X have ime coordinaes which are independen and uniformly disribued over he inerval [0, 1]. Assume balls of volume 1 cenered a he poins of X arrive sequenially in an order deermined by he ime coordinaes, and assume as before ha each ball is packed or discarded according o wheher or no i overlaps a previously packed ball. Le ξ(x; X ) be eiher 1 or 0 depending on wheher he ball cenered a x is packed or discarded. Consider he re-scaled packing funcional ξ (x; X ) = ξ( 1/d x; 1/d X ), where 1/d x denoes scalar muliplicaion of x bu no he mark associaed wih x and where balls cenered a poins of 1/d X have volume one. The random measure N() µ ξ := i=1 ξ (X i ; {X i } N() i=1 ) δ X i, is called he random sequenial packing measure induced by balls wih ceners arising from κ. The convergence of he finie dimensional disribuions of he packing measures µ ξ is esablished in [, 4]. ξ is exponenially sabilizing [10, ] and for any f B c ([0, 1] d ) and κ uniform, he variance of f, µ ξ is bounded below by a posiive muliple of [4], showing ha f, µξ saisfies a CLT wih an O((log ) d 1/2 ) rae of convergence. I follows easily from he sabilizaion analysis of [10] ha many varians of he above basic packing model saisfy similar raes of convergence in he CLT. For example, he number of balls acceped in he cooperaive sequenial adsorpion models and he monolayer ballisic deposiion models of [10] boh saisfy he CLT wih an O((log ) d 1/2 ) rae of convergence. commen applies for he number of seeds acceped in he spaial birh-growh models [10]. The same 2.4 Independence number, off-line packing An independen se of verices in a graph G is a se of verices in G, no wo of which are conneced by an edge. The independence number of G, which we denoe β(g), is defined o be he maximum cardinaliy of all independen ses of verices in G. For r > 0, and for finie or counable X R d, le G(X, r) denoe he geomeric graph wih 7

8 verex se X and wih edges beween each pair of verices disan a mos r apar. Then he independence number β(g(x, r)) is he maximum number of disjoin closed balls of radius r/2 ha can be cenered a poins of X ; i is an off-line version of he packing funcionals considered in he previous secion. Le b > 0 be a consan, and consider he graph G(P, b 1/d ) (or equivalenly, G( 1/d P, b)). Random geomeric graphs of his ype are he subjec of [7], alhough independence number is considered only briefly here (on page 15). A law of large numbers for he independence number is described in Theorem 2.7 (iv) of [11]. For µ > 0, le H µ denoe a homogeneous Poisson poin process of inensiy µ on R d, and le Hµ 0 be he poin process H µ wih a poin insered a he origin. As on page 189 of [7], le c be he infimum of all µ such ha he origin has a non-zero probabiliy of being in an infinie componen of G(H µ, 1). If b d κ < c, we can use Theorem 1.1 o obain a cenral limi heorem for he independence number β(g(p, b 1/d )). We only skech he proof. The graph G(P, b 1/d ) is isomorphic o G(b 1 1/d P, 1) and he poin process b 1 1/d P is dominaed by H b d κ (in he sense of [7], page 189). By exponenial decay for subcriical coninuum percolaion (Lemma 10.2 of [7]) he probabiliy ha he componen of G(H 0 b d κ, 1) conaining he origin includes a poin disan more han r from he origin decays exponenially in r, and one can deduce exponenial sabilizaion from his. Proof of Theorem A CLT for dependency graphs We shall prove Theorem 1.1 by showing ha exponenial sabilizaion implies ha a modificaion of f, µ ξ has a dependency graph srucure, whose definiion we now recall (see e.g. Chaper 2 of [7]). Le X α, α V, be a collecion of random variables. The graph G := (V, E) is a dependency graph for X α, α V, if for any pair of disjoin ses A 1, A 2 V such ha no edge in E has one endpoin in A 1 and he oher in A 2, he sigma-fields σ{x α, α A 1 }, and σ{x α, α A 2 }, are muually independen. Le D denoe he maximal degree of he dependency graph. I is well known ha sums of random variables indexed by he verices of a dependency graph admi raes of convergence o a normal. The raes of Baldi and Rino [2] and hose in Penrose [7] are paricularly useful; Avram and Bersimas [1] use he former o obain rae resuls for he oal 8

9 edge lengh of he neares neighbor, Voronoi, and Delaunay graphs. In many cases, he following heorem of Chen and Shao [6] provides superior rae resuls. We shall apply his resul when p =. For any random variable X and any p > 0, le X p = (E [ X p ]) 1/p. Lemma.1 (see Thm 2.7 of [6]) Le 2 < p. Le W i, i V, be random variables indexed by he verices of a dependency graph. Le W = i V W i. Assume ha E [W 2 ] = 1, E [W i ] = 0, and W i p θ for all i V and for some θ > 0. Then sup P [W ] Φ() 75D 5(p 1) V θ p. (.1).2 Auxiliary lemmas To prepare for he proof of Theorem 1.1 we will need some auxiliary lemmas. We assume hroughou ha A [0, 1] d, bu all of our resuls can be easily modified o rea he case of arbirary compac ses A R d. Throughou, C denoes a generic consan depending possibly on d, ξ, and κ and whose value may vary a each occurrence. We assume > 1 hroughou. Le α > 0 be a consan o be chosen laer. Given > 0, le s := 1/d α log, and cover [0, 1) d by cubes of side s of he form d i=1 [j is, (j i + 1)s ), wih all j i Z. Le he cubes in he covering be denoed Q 1, Q 2,..., Q V, where V := V () is he number of cubes in he covering, i.e. V () := s 1 d = 1/d /(α log ) d. For all 1 i V (), he number of poins in P Q i is a Poisson random variable N i := N(τ i ), where τ i := Q i κ(x)dx. Assuming τ i > 0, choose an ordering on he poins of P Q i uniformly a random from all (N i )! possible such orderings. Use his ordering o lis he poins as X i,1,..., X i,ni, where condiional on he value of N i, he random variables X i,j, j = 1, 2,... are i.i.d. on Q i wih a densiy κ i ( ) := κ( )/ Q i κ(x)dx. Thus we have he represenaion P = V () i=1 {X i,j} Ni. For all 1 i V (), le P i := P \ {X i,j } Ni and noe ha P i is a Poisson poin process on [0, 1] d wih inensiy densiy κ on [0, 1] d \ Q i and inensiy zero on Q i. We show ha he condiion (1.1), which bounds he momens of he value of ξ a poins insered ino P, also yields bounds on E [ ξ (X i,j ; P ) 1 j Ni p ]. More precisely, we have Lemma.2 Le p > 0. If (1.1) holds, hen here is a consan C such ha for all > 1 and all 1 i V () E [ ξ (X i,j ; P ) 1 j Ni p ] C(log ) d. (.2) 9

10 Proof. If N i = n, hen denoe {X i,1,..., X i,ni } by X n. We have by definiion E [ ξ (X i,j ; P ) 1 j Ni p ] = n=j Q i E [ ξ (x; X n 1 P i ) p ]κ i (x)dx P [N i = n], where he expecaion on he righ hand side is wih respec o X n 1 and P i. The above is bounded by = τ i m=0 τ i n=1 n 1 i E [ ξ(x; X n 1 P i ) p ]κ i (x)dx e τi τ Q i (n 1)! Q i E [ ξ (x; P ) p P Q i = m] κ i (x)dx P [ P Q i = m] = τ i Q i E [ ξ (x; P x) p ]κ i (x)dx cons. τ i, where he las inequaliy follows by (1.1). Since τ i = Q i κ(x)dx κ (α log ) d, his shows (.2). Fix 1 i V. For all j = 1, 2,... we define ξ j := ξ i,j := ξ (X i,j ; P ) when 1 j N i and oherwise we se ξ j = 0. Lemma. If (1.1) holds for some p >, hen ξ j C(log ) 4d. Proof. Clearly, wih N := N i and τ := τ i, ( ) ξ j = ξ j 1 N τ k τ<n 2 k+1 τ ξ j 1 2 k τ<n 2 k+1 τ + ξ j 1 N τ. Since a.s. only finiely many summands in he double sum are non-zero, by subaddiiviy of he norm, he above is bounded by τ ξ j 1 2 k τ<n 2 k+1 τ + ξ j 1 N τ 10

11 2 k+1 τ τ ξ j 1 N 2 k τ + ξ j 1 N τ 2 k+1 τ ξ j 1 N 2 k τ + Hölder s inequaliy yields for all 1 j 2 k+1 τ and any 0 < δ < 1/9: and herefore replacing δ wih δ gives τ ξ j 1 N τ. (.) ξ j 1 N 2 k τ ξ j +δ (P [N 2 k τ]) δ/(+δ) ξ j 1 N 2 k τ ξ j +δ (P [N 2 k τ]) δ/10. (.4) Now by (.2) we have ξ j +δ C(log ) d/(+δ). (.5) Subsiuing (.4) and (.5) ino (.) we obain ξ j C(log ) d/(+δ) τ2 k+1 (P [N 2 k τ]) δ/10 + Now for τ > 1 we have τ2 k+1 (P [N 2 k τ]) δ/10 τ2 k+1 (P [N 2 k ]) δ/10 Cτ whereas for 0 < τ < 1 we have log 2 1 τ +2 C + τ2 k+1 (P [N 2 k τ]) δ/10 τ2 k+1 + τ ξ j 1 N τ. (.6) k= log 2 1 τ + τ2 k+1 (P [N 2 k τ]) δ/10 k= log 2 1 τ + τ2 k+1 (exp( 2 k 1 τ k)) δ/10 C, where he penulimae inequaliy follows from bounds for he ail of a Poisson (see e.g. (1.12) in [7]). Thus he firs sum on he righ hand side of (.6) is a mos C(log ) 4d/ since τ C(log ) d. Since ξ j C(log ) d/ we find ha (.6) implies ξ j C(log ) 4d/, (.7) which concludes he proof of Lemma.. 11

12 . Conclusion of proof of Theorem 1.1 Throughou his secion, we fix f B(A) and se T := f, µ ξ. For all 1 i V and all j = 1, 2,..., le R i,j denoe he radius of sabilizaion of ξ a 1/d X i,j if 1 j N i and le R i,j be zero oherwise. E [ V Le E i,j := {R i,j α log }. Then by sandard Palm heory (e.g. Theorem 1.6 in [7]) ] Ni i=1 1 Ei,j c = P [R(x, ) > α log ]κ( 1/d x)dx C by exponenial sa- [0, 1/d ] d bilizaion if α is large enough. Le E := V i=1 E i,j and noe ha P [E c ] C. Recalling he represenaion P = V () i=1 {X i,j} N i, we have T = V () N i i=1 ξ (X i,j ; P ) f(x i,j ). To obain raes of normal approximaion for T, i will be be convenien o consider a closely relaed sum enjoying more independence srucure, namely For all 1 i V () define T := V () N i i=1 ξ (X i,j ; P ) 1 Ei,j f(x i,j ). N i S i := S Qi := (VarT ) 1/2 ξ (X i,j ; P ) 1 Ei,j f(x i,j ) and pu S := (VarT ) 1/2 (T E T ) = V () i=1 (S i E S i ). Clearly VarS = E S 2 = 1. We define a graph G := (V, E ) as follows. The se V consiss of he subcubes Q 1,..., Q V () and edges (Q i, Q j ) belong o E if d(q i, Q j ) 2α 1/d log, where d(q i, Q j ) := inf{ x y : x Q i, y Q j }. By definiion of exponenial sabilizaion, we noe ha if A 1 and A 2 are disjoin collecions of cubes in V such ha no edge in E has one endpoin in A 1 and one endpoin in A 2, hen he random variables {S Qi, Q i A 1 } and {S Qj, Q j A 2 } are independen. Thus G is a dependency graph. To prepare for an applicaion of Lemma.1, we make four observaions: (i) V () := V = 1/d /(α log ) d. (ii) Since he number of cubes in Q 1,..., Q V disan a mos 2α 1/d log from a given cube is bounded by 5 d, i follows ha he maximal degree D saisfies D := D 5 d. (iii) The definiions of S i and ξ i,j and Lemma. ell us ha for all 1 i V () E [ S i ] C(VarT ) /2 E ξ i,j C(VarT ) /2 (log ) 4d. 12

13 (iv) Var[T ] is close o Var[T ] for large. We require a few esimaions o show his. Noe ha T T = 0 excep possibly on he se E c which has probabiliy less han C. Lemma., along wih Minkowski s inequaliy, yields he upper bound V () N i E ξ (X i,j ; P ) CV () (log ) 4d C 4. (.8) i=1 Thus Hölder s inequaliy implies ha E [ T T 2 ] E [ T T 2 ] 1 E c 4E [(T 2 + T 2 ) 1 E c ] 2/ V () N i 8 E ξ (X i,j ; P ) (P [E]) c 1/ C 8 (.9) i=1 and hus E [ T T ] C 4. (.10) Addiionally, (.8) and Jensen s inequaliy yield 2 V () N i E ξ (X i,j ; P ) C. (.11) i=1 Since Var[T ] = Var[T ] + Var(T T ) + 2Cov(T, T T ), by (.9), (.11) and he Cauchy-Schwarz inequaliy we obain Var[T ] Var[T ] C 2. (.12) Given he four observaions (i)-(iv), we are now ready o apply Lemma.1 o prove Theorem 1.1. Trivially, (1.2) holds for large enough when Var[T ] < 1, and so wihou loss of generaliy we now assume Var[T ] 1. To esablish he rae of convergence (1.2) in his case, we apply he bound (.1) o W i := S i E S i, 1 i V, wih p = and wih θ := C(VarT ) 1/2 (log ) 4d/. Our choice of θ is applicable because of observaion (iii). E [( V i=1 W i) 2 ] = 1. Wih S = V i=1 W i, Lemma.1 yields We clearly have E [W i ] = 0 and sup P [S ] Φ() C 1/d α log d (VarT ) /2 (log ) 4d 1

14 C(VarT ) /2 (log ) d, (.1) where he las line makes use of he fac ha Var[T ] Var[T ]/2, which follows from (.12). Now if β > 0 is a consan and Z any random variable hen by (.1) we have for all R P [Z ] P [S + β] + P [ Z S β] Φ( + β) + C(VarT ) /2 (log ) d + P [ Z S β] Φ() + Cβ + C(VarT ) /2 (log ) d + P [ Z S β] by he Lipschiz propery of Φ. Similarly for all R In oher words P [Z ] Φ() Cβ C(VarT ) /2 (log ) d P [ Z S β]. sup P [Z ] Φ() Cβ + C(VarT ) /2 (log ) d + P [ Z S β]. (.14) Now by definiion of S, (VarT ) 1/2 (T E T ) S = (VarT ) 1/2 {(T E T ) (T E T )} (VarT ) 1/2 { T T + E [ T T ]} which by (.10) is bounded by C 4 excep possibly on he se E c which has probabiliy less han C. Thus by (.14) wih Z = (VarT ) 1/2 (T E T ) and β = C 4 sup P [(VarT ) 1/2 (T E T ) ] Φ() C(VarT ) /2 (log ) d + C 4 + C. Moreover, by he riangle inequaliy sup P [(VarT ) 1/2 (T E T ) ] Φ() [ sup P (VarT ) 1/2 (T E T ) ( VarT ] ( VarT ) 1/2 Φ ( VarT ) VarT ) 1/2 + ( + sup Φ ( VarT ) VarT ) 1/2 Φ(). Since for all s, we have Φ(s) Φ() ( s) max s u φ(u) where φ denoes he sandard normal densiy, and since by (.12) here is a consan 0 < C < such ha for all > 0 and all R (VarT VarT ) 1/2 VarT VarT 1 C 2 14

15 we ge Thus, sup ( Φ ( VarT ) VarT ) 1/2 Φ() C sup (( ) ( 2 max φ(u) u [ C/ 2, +C/ 2 ] )) C 2. sup P [(VarT ) 1/2 (T E T ) ] Φ() C(VarT ) /2 (log ) d + C 2. (.15) Finally we asser ha VarT = O((log ) 8d/ ). (.16) To see his, observe ha T is he sum of V () random variables, which by Jensen s inequaliy and Lemma. each have a second momen bounded by a consan muliple of (log ) 8d/. Thus he variance of each of he V () random variables is also bounded by a consan muliple of (log ) 8d/. Moreover, he covariance of any pair of he V () random variables is zero when he indices of he random variables correspond o non-adjacen sub-cubes. For adjacen sub-cubes, he covariance is also bounded by a consan muliple of (log ) 8d/. This shows ha VarT = O((log )8d/ ), and combined wih (.12) his yields (.16). By (.16), in (.15) he firs erm in he righ hand side dominaes, hus yielding he desired bound (1.2), and he proof of Theorem 1.1 is complee. Acknowledgmens. We began his work while visiing he Insiue for Mahemaical Sciences a he Naional Universiy of Singapore, and coninued i while visiing he Isaac Newon Insiue for Mahemaical Sciences a Cambridge. We hank boh insiuions for heir hospialiy. References [1] F. Avram and D. Bersimas (199), On cenral limi heorems in geomerical probabiliy. Ann. Appl. Probab., [2] P. Baldi and Y. Rino (1989), Asympoic normaliy of some graph relaed saisics. J. Appl. Probab. 26, [] Yu. Baryshnikov and J.E. Yukich (200), Gaussian fields and random packing. J. Sais. Phys. 111,

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