YINGHUA DONG AND GENNADY SAMORODNITSKY

CONTACT DISTRIBUTION IN A THINNED BOOLEAN MODEL WITH POWER LAW RADII YINGHUA DONG AND GENNADY SAMORODNITSKY Abstact. We conside a weighted stationay spheical Boolean model in R d. Assuming that the adii of the balls in the Boolean model have egulaly vaying tails, we establish the asymptotic behaviou of the tail of the contact distibution of the thinned gem-gain model unde 4 diffeent thinning pocedues of the oiginal model. 1. Intoduction We conside had-coe global thinning of a stationay spheical Boolean model in R d, constucted as follows. Let Φ be a Poisson andom measue on R d,, with mean measue 1.1 mdx, d, dw = λ dx Gd, dw. Hee λ > is the spatial intensity, and G is a pobability law on,,. Let X n, R n, W n, n = 1, 2... be a measuable enumeation of the points of Φ. We view X n R d as the cente of the nth ball, and R n its adius. In the sequel we will use the notation B x fo a closed ball of adius > centeed at x R d, so the nth point of Φ coesponds to closed ball B Rn X n. The last component, W n, is the weight of the nth ball, and it will be used below in esolving collisions between balls. Let F = G, be the law of the adius making a spatial Poissonian point. We will assume that 1.2 d F d <. It is well known that, unde this assumption, with pobability 1, a ealization of the andom field Φ has the popety that only finitely many balls of the type B Rn X n intesect any compact set in R d. This implies that the union 1.3 D = B Rn X n n=1 is a andom closed subset of R d. We efe the eade to Stoyan et al. 1995 fo this fact, and fo a geneal efeence on Boolean and elated models. It is common to efe to a andom configuation of the type we have constucted as a gem-gain model; such a model does not need to involve a spatial Poisson andom measue o spheical shapes. 1991 Mathematics Subject Classification. Pimay 6D5. Key wods and phases. Boolean model, had-coe thinning, powe tails, contact distibution, empty space. Dong s eseach was patially suppoted by Jiangsu Planned Pojects fo Postdoctoal Reseach Funds No. 15153A, the Statistical Scientific Reseach Pogam of National Bueau of Statistics of China No. 215LY83 and the foundation of Nanjing Univesity of Infomation Science and Technology No. 214x26. Samoodnitsky s eseach was patially suppoted by the ARO gant W911NF-12-1385 and by the NSF gant DMS-156783 at Conell Univesity. 1

2 YINGHUA DONG AND GENNADY SAMORODNITSKY In the Boolean model above each X n is a gem, and the coesponding closed ball B Rn is its gain. The set D in 1.3 is the gain cove of the space. Some of the balls B Rn X n in the Boolean model will ovelap. In odet to obtain a had-coe gem-gain model, i.e. a configuation in which no two gains ovelap, it is possible to thin the Boolean model, by emoving at least one ball in each pai of balls involved in an ovelap. We will follow the global thinning pocedue intoduced by Mansson and Rudemo 22. This is whee the weight component W n of the nth ball is used. Infomally, fo evey pai of diffeent balls, B Rn X n and B Rm X m with a non-empty intesection, the ball B Rn X n gets deleted if W n W m ; this pocedue deletes both balls if W n = W m. To be a bit moe fomal, we use the notation boowed fom Kuonen and Leskelä 213: let 1.4 N x,,w = { x,, w R d R + R + \{x,, w} : B x B x } ; we view this set as the collection of centes, adii and weights of balls that could, potentially, intesect a efeence ball B x with weight w. Then the thinned Boolean model we ae consideing is given by 1.5 Φ th = { x,, w Φ : w > w, fo all x,, w Φ N x,,w }. By constuction, all the emaining gains balls in the thinned andom field Φ th ae disjoint. The coesponding gain cove can be witten in the fom 1.6 D th = B x. x,,w Φ th The thinning pocedue we ae using is sometimes efeed to as the Matén type II constuction. An discussion of diffeent Matén type constuctions and thei extensions was given in Nguyen and Baccelli 213. This model was studied in Mansson and Rudemo 22 and Andeson et al. 26. Ou inspiation fo the pesent wok came fom the pape of Kuonen and Leskelä 213, and we efe the eade to this pape fo an illuminating discussion of the impotance and applications of had-coe gem-gain models. Specifically, the latte pape consides the case of powe law gain sizes; in ou notation we can descibe this setup as follows. Recalling that we denote by F the maginal distibution of the pobability measue G in 1.1 coesponding to the andom adius of a Poisson ball, the powe law distibution of the gain sizes in Kuonen and Leskelä 213 is the assumption of egula vaiation of the tail 1.7 F := 1 F = α L, whee L is a slowly vaying function and α > d; the latte estiction assues that the integability condition 1.2 holds. We efe the eade to Resnick 27 fo infomation on egula vaying tails. Unde the assumption 1.7 of egula vaiation, Kuonen and Leskelä 213 discoveed appeaance of powe-like decay of the covaiance function of the thinned gain cove 1.6, unde thee out of fou choices of the joint law G in 1.1 they consideed; we will etun to these choices in a moment. In this pape we ae inteested in the contact distibution fo the thinned Boolean model descibed above. It is a pobability law H on, whose complementay c.d.f. is defined by 1.8 H = P B D th = / D th, >. The contact distibution can, of couse, be defined fo any gem-gain model. It diffes only by a possible atom at zeo fom the empty space function, a pobability law on [, defined by 1.9 He = P B D th =,.

CONTACT DISTRIBUTION IN A THINNED MODEL 3 Contact distibutions ae impotant chaacteistics of gem-gain models; a suvey on the topic is in Hug et al. 22. Explicit fomulas fo the contact distibutions ae mostly available only fo Poisson-based models such as Poisson cluste models. Ou goal in this pape is to undestand the tail behaviou of the contact distibution fo the thinned Boolean model with a powe law distibution of the gain sizes. Specifically, we ae inteested in answeing the question whethe a powe law distibution of the gain sizes esults in a powe law behaviou of the contact distibution fo the thinned Boolean model. Notice that fo the oiginal Boolean model with the gain cove 1.3 the tail of the contact distibution decays, obviously, exponentially fast egadless of the distibution of the adius of a ball. It tuns out that cetain choices of the joint law G of the adius of a ball and its weight lead to appeaance of a powe law-like decay of the contact distibution, while othe choices do not. One possible choice of the the law G in 1.1 is given by setting W n = R n a.s. fo all n, so that G is concentated on the diagonal = w of,,. With this choice of G, balls with a lage adius have a lage weight. We efe to this situation as the case of heavy lage balls. Anothe possible choice of G is given by setting W n = 1/R n a.s. fo all n. With this choice of G, balls with a smalle adius have a lage weight. A thid possible choice of the law G is to make it a poduct law, and to make the maginal law of the weights continuous e.g. standad unifom. That is, the weights ae independent of the adii of the balls. Finally, one could make the weights of the balls constant e.g. W n = 1 a.s. fo all n. In this case, only isolated balls in the oiginal Boolean model i.e. the balls that do not ovelap with any othe ball stay in the thinned gain-gem model Φ th. The latte thinning mechanism is known as the Matén type I constuction. It is inteesting that, as it was shown in Kuonen and Leskelä 213, when the adii of the balls ae egulaly vaying as in 1.7, the covaiance function of the thinned gain cove 1.6 has a powe-like decay unde all of the above thinning mechanisms apat fom the case of heavy small balls. In a cetain sense the above situation is peseved when one is inteested in the tail of the contact distibution. In Section 2 we show that this tail has a powe-like decay in all cases apat fom the case of heavy small balls. Inteestingly, the powe of the ate of decay is diffeent fo the thee thinning mechanisms consideed hee. Finally, In Section 3, on the othe hand, we show that in the situation when a ball with a lage adius has a smalle weight than a ball with a small adius, the tail of the contact distibution decays exponentially fast. 2. Powe law of the contact distibution We stat this section with showing that, in the case when a ball with a lage adius in a Boolean model has a lage weight than a ball with a smalle adius, the contact distibution of the thinned model has a tail with powe-like decay. Theoem 2.1. Assume that the distibution of the adii of the balls in the Boolean model satisfies 1.7 with α > d. If lage balls have lage weights, then the contact distibution of the thinned gem-gain model satisfies 2.1 < lim inf H d F 2 lim sup H d F 2 <. Poof. Thoughout the poof, we may and will wok with the tail of the empty space function 1.9 instead of the tail of the contact distibution. Futhe, we denote by c a finite positive constant whose value is not impotant and that may change fom one appeaance to the next. We will also intoduce a notational simplification. The Poisson andom measue Φ is a measue in the d + 2- dimensional space R d,,, but in the pesent context the weight coodinate is a

4 YINGHUA DONG AND GENNADY SAMORODNITSKY function of the adius coodinate, so it is simple to view Φ as a measue in the d + 1-dimensional space R d,, descibed by the location of the cente of a ball and its adius. We will use the appopiate notation thoughout the poof. We stat with poving the lowe bound in 2.1. We will constuct a scenaio unde which the ball B does not intesect Φ th. The idea of the constuction is that a single ball with a lage adius in Φ eliminates all the othe balls in Φ that intesect B, and then anothe ball in Φ of an even lage adius eliminates the fist ball of a lage adius, but does not itself intesect B. That will leave B disjoint fom Φ th. The two lage balls will have centes in sets of sizes popotional to, and also adii of the size popotional to, which explains the ode of magnitude of the tail in 2.1. Fo > we conside thee disjoint subsets of R d, : 2.2 A 1 = { x, t : t x + }, 2.3 A 2 = { x, t : max, x t < x + }, 2.4 A 3 = { x, t : x t < }. Notice that only those balls B Rn X n in Φ fo which X n, R n A 1 A 2 A 3 intesect B. contains the entie ball B as a will be most impotant fo us in poving the lowe bound in 2.1. Conside Futhemoe, any ball B Rn X n in Φ fo which X n, R n A 1 subset. The set A 1 the event { B = Φ A 1 } = 1, Φ A 2 =. On the event B we can define a andom vecto X, R A 1 coesponding to the location of the cente and the adius of the single ball in Φ fo which that pai is in the set A 1. We extend the definition of X, R to the outside of the event B in an abitay measuable way e.g. define it on B c to be the pai, 1. Clealy, this vecto has the law P X, R B = Leb F Leb F A 1 ove A 1. Hee Leb is the d-dimensional Lebesgue measue. Note that 2.5 He = P B D th = P B ˆB, whee ˆB = { thee is a Φ-ball B Rn X n with X n, R n A 1 A 2 A 3 and R n > R that intesects B R X. } Let v d be the volume of the unit ball in R d. By switching to the spheical coodinates we see that fo the lage, 2.6 E [ Φ A 1 ] =λdvd x d 1 F x + dx =λdv d d t d 1 F t + 1 dt c d F. c

CONTACT DISTRIBUTION IN A THINNED MODEL 5 On the last step we used the Potte bounds fo egulaly vaying functions; see Resnick 27. Theefoe, fo lage, P B E [ Φ A 1 ] 2.7 c d F. Similaly, E [ Φ A 2 ] c d F, as. Next, fo y, w A 1 we denote A,y,w = { x, t A 1 A 2 A 3 c : t > w, the ball B t x intesects the ball B w y }. Then, since a Poisson andom measue assigns independent values to disjoint sets, P ˆB 1 B = Leb F A 1 P ΦA,y,w > dy F dw y,w A 1 1 5.5 { 1 exp ma,y,w Leb F A 1 } dy F dw B 3 \B 2 5 because, obviously, It follows fom 2.6 that B3 \ B 2 5, 5.5 A 1. Leb F [ B 3 \ B 2 5, 5.5 ] Leb F A 1 fo all lage. Theefoe, the lowe bound in 2.1 will follow fom 2.5 and 2.7 once we show that thee is a constant c such that fo all lage enough 2.8 ma,y,w c d F fo all y, w B 3 \ B 2 5, 5.5. To this end, fo such a pai y, w conside the point ỹ = y y y + w Rd, and the ball B ỹ. Let z. Note that the distance fom the point ỹ + z B ỹ to the ball B w y does not exceed z, while the distance fom that same point to ball B is geate than y + w > w +. Taking into account the bounds on w we have chosen, we see that any ball centeed at a point ỹ + z B ỹ with a adius t 5.5, 6 will intesect the ball B w y but not the ball B. We conclude that fo a pai y, w as above, implying that c A,y,w { x, t : x B ỹ, t 5.5, 6 }, m A,y,w c d F 5.5 F 6 c d F as, by the egula vaiation. This poves 2.8. Now we switch to poving the uppe bound in 2.1. Let K > be a fixed numbe to be specified momentaily. Denote A 4 K = { x, t : max/k, x t < x + }. The same agument using egula vaiation and the Potte bounds as in 2.6 shows that fo lage, 2.9 E [ Φ A 4 K ] c d F

6 YINGHUA DONG AND GENNADY SAMORODNITSKY with a K-dependent constant c. This bound, togethe with 2.6, tells us that fo lage, P Φ A 1 2 c d 2, F P Φ A 4 K 2 c d 2 F, P Φ A 1 1, Φ A 4 K 1 c d 2 F. Theefoe, the uppe bound in 2.1 will follow once we pove the following 3 statements. Fo lage, 2.1 P B D th =, Φ A 1 = 1 c d 2 F, 2.11 P B D th =, Φ A 4 K = 1 c d F 2, 2.12 P B D th =, Φ A 1 = Φ A 4 K = c d 2 F. We will see that 2.1 and 2.11 hold fo any K >. We will specify K when we pove 2.12. We stat with poving 2.1. Fo the event in that pobability to occu, the only Φ-ball in A 1 must ovelap with anothe Φ-ball, of a lage adius, and lying outside of A 1. Since estictions of a Poisson measue to disjoint sets ae independent, and since the only Φ-ball in A 1 has a adius of at least, the pobability in 2.1 is bounded fom above by P Φ A 1 = 1 sup P ΦA 5 s >, s whee 2.13 A 5 s = { x, t R d, : t > s, the ball B t x intesects the ball B s }, the cente of the ball of adius s being ielevant due to the stationaity. It is elementay that fo lage s, by the egula vaiation of F and Kaamata s theoem on integation of egulaly vaying functions see e.g. Resnick 27, 2.14 m A 5 s =c x d 1 F s x s dx =c2s d F s + c x d 1 F x s dx cs d F s. Theefoe, fo lage s, P ΦA 5 s > cs d F s, and 2.1 follows fom 2.6. Clealy, all the ingedients involved in the poof of 2.1 ae also available fo the poof of 2.11, so we only need to pove 2.12. Now we explain how to choose K. It will be chosen togethe with seveal othe constants. Choose sequentially positive eal numbes < θ < d/α and < τ < θα d, a positive intege I > 2α d/τ 1 and, finally, K > 2 I+1. We stat with consideing the concentic balls B 2 i, i =, 1,..., I. Fo i =, 1,..., I, let Then M i = sup { R n : B Rn X n is an Φ-ball, X n + R n < 2 i}. P M i 2 i { θ {x, = exp m t : x + t < 2 i, t > 2 i θ} }. 2s

CONTACT DISTRIBUTION IN A THINNED MODEL 7 Futhe, since θ < 1, {x, m t : x + t <, t > θ } {x, m t : x < /2, θ < t /2 } =c d F θ F /2 c d F θ as. By the choice of θ we see that P M i 2 i θ = o d 2, F i =, 1,..., I, and so 2.12 will follow once we pove that 2.15 lim sup P B D th =, Φ A 1 = Φ A 4 K =, M i > 2 i θ, i =, 1,..., I d F 2 <. Conside the events { H i = the Φ-ball fully inside B 2 i of the lagest adius, } is eliminated by an Φ-ball not fully inside B 2 i, i =, 1,..., I. Note that, on the event Hi c, the lagest Φ-ball fully inside B 2 i stays in the thinned pocess, hence B D th. Theefoe, in ode to pove 2.15, it is enough to pove that lim sup d 2P F {Φ A 1 = Φ A 4 K =, M i > 2 i } θ 2.16, i =, 1,..., I H... H I <. Conside fist the pobability { P Φ A 1 = Φ A 4 K =, M i > 2 i } θ, i =, 1,..., I H I { P Φ A 1 = Φ A 4 K =, M I > 2 I } θ H I. On the latte event, we can define a andom vecto XI, R I as the cente and the adius of the lagest Φ-ball fully within B 2 I. Note that R I > 2 I θ. The andom vecto XI, R I is detemined by the field Φ on the set {x, t : x + t < 2 I }, and the coesponding Φ-ball can only be eliminated by the Φ-balls in the complement of that set. Since estictions of a Poisson measue to disjoint sets ae independent, we conclude, in the notation of 2.13 that fo lage, { P Φ A 1 = Φ A 4 K =, M I > 2 I } θ H I sup P ΦA 5 s > s 2 I θ c 2 I θd F 2 I θ c τ, whee on the last two steps we used 2.14, the choice of τ and the egula vaiation of F.

8 YINGHUA DONG AND GENNADY SAMORODNITSKY Next we conside the pobability P Φ A 1 = Φ A 4 K =, M i > 2 i } θ, i =, 1,..., I H I 1 H I P Φ A 1 = Φ A 4 K =, M i > 2 i } θ, i = I 1, I H I 1 H I. Note that the condition Φ A 1 4 = Φ A K = in the above event means that the lagest Φ-ball completely within B 2 I could only be eliminated by a Φ-ball centeed at a point whose nom is in the ange 2 I ± /K, while the lagest Φ-ball completely within B 2 I 1 could only be eliminated by a Φ-ball centeed at a point whose nom is in the ange 2 I 1 ± /K. These two anges ae disjoint by the choice of K. We use, once again, the fact that estictions of a Poisson measue to disjoint sets ae independent, and an agument as above gives us P Φ A 1 = Φ A 4 K =, M i > 2 i θ, i = I 1, I } H I 1 H I c τ 2. Poceeding in the same manne, we finally obtain { P Φ A 1 = Φ A 4 K =, M i > 2 i } θ, i =, 1,..., I H... H I c τ I+1 fo lage. By the choice of I, we see that 2.16 follows. Now we conside the case of isolated balls emaining. Once again, the contact distibution has a powe-like decaying tail, but the coesponding powe is diffeent fom the powe obtained in Theoem 2.1. This is, pehaps, not supising, since keeping only isolated balls esults in fewest balls emaining in the thinned model, hence lage open space. Theoem 2.2. Assume that the distibution of the adii of the balls in the Boolean model satisfies 1.7 with α > d. If only isolated balls ae kept in the thinned model, then the contact distibution of the thinned gem-gain model satisfies 2.17 < lim inf H d F lim sup H d F <. Poof. We use the same conventions as in the poof of Theoem 2.1. In paticula, we wok with the tail of the empty space function, and view the Poisson andom measue Φ as a measue in the d + 1-dimensional space R d,. Once again, we stat with a lowe bound. One scenaio unde which the ball B is disjoint fom the gain cove in the thinned model is existence of a Φ-ball that coves the entie ball B plus existence of a Φ-ball that is entiely within the ball B. Since as, we conclude by 2.6 that m { x, t : B t x B } P B D th = 1 o1 P ΦA 1 1 E ΦA 1 c d F. This poves the lowe bound in 2.17. The agument fo the uppe bound in 2.17 is based on seveal facts. Fist of all, since the thinned andom filed Φ th is a.s. non-empty, fo any ε > and lage enough a >, 2.18 P thee is B v x Φ th with x ε and v a >.

CONTACT DISTRIBUTION IN A THINNED MODEL 9 Secondly, thee is a constant c > such that fo any < a thee exist at least [c d /a d ] closed balls of adius a completely within B, such that the Euclidian distance between any two diffeent balls is at least a. This fact can be easily veified by consideing a egula gid of size a inside B. Let M be the lagest adius of a Φ-ball intesecting B. Clealy, fo any t >, whee P M > t = 1 e ma6,t, A 6,t = { x, s R d, : s > t, the ball B s x intesects the ball B }. An agument simila to the one in 2.14 shows that 2.19 m A 6,t c d F t, with a simila lowe bound, but with a diffeent constant c. Wite P B D th = P M > + P B D th = M = t F M dt, whee F M is the law of M. It follows fom 2.19 that we have to pove that 2.2 lim sup P B D th = M = t F M dt d F <. The popety that estictions of a Poisson measue to disjoint sets ae independent tells us that, given that M = t, the andom measue Φ esticted to the set A 6 t = {x, s : s < t} is still a Poisson andom measue on that set with the same mean measue m, esticted to that set. Take a > such that 2.18 holds, and choose ε = a. Let < p < 1 be the coesponding value of the pobability in 2.18. Conside a < t <. By 2.18, thee ae [c d /t d ] closed balls of adius t completely within B, such that the Euclidian distance between any two diffeent balls is at least t. Fo each one of these [c d /t d ] balls, with pobability at least p, thee is an isolated Φ-ball with a cente in it, and adius not exceeding t. The events that such Φ-balls exist ae independent, and pesence of such a Φ-ball guaantees that B D th. Theefoe, fo any t > a, It is clea that Futhemoe, P B D th = M = t 1 p [cd /t d] 1 p 1 1 p cd /t d. c a a P M a e cd = o d F. P B D th = M = t F M dt e d /ct d F M dt c d e d /ct d t d+1 FM t dt c 2d e d /ct d t d+1 F t dt 1 =c d e 1/csd s d+1 F s ds

1 YINGHUA DONG AND GENNADY SAMORODNITSKY 1 c d F e 1/csd s α+d+1 ds as by the egula vaiation of F and the Potte bounds. This completes the poof of 2.2 and, hence, of the uppe bound in the theoem. Finally, we conside the case when the weights ae independent of the adii of the balls. Theoem 2.3. Assume that the distibution of the adii of the balls in the Boolean model satisfies 1.7 with α > d. If the weight of a ball in the model is independent of its adius and has a continuous distibution, then the contact distibution of the thinned gem-gain model satisfies 2.21 < lim inf H F lim sup H F <. Poof. The stuctue of the agument is simila to that in Theoem 2.1. We also follow the conventions in the poof of Theoem 2.1 by woking with the tail of the empty space function. Howeve, since in this case the weight of a Φ-ball is not a function of its adius, we have to view the Poisson andom measue Φ as a measue in the full d + 2-dimensional space R d,,. As befoe, we stat with poving the lowe bound in 2.21. The scenaio we will use is simila to the scenaio we used to pove the lowe bound in Theoem 2.1. Specifically, in this scenaio thee is a single ball of a lage adius and lage weight that eliminates all the othe Φ-balls that intesect B, and then anothe ball in Φ of an even lage weight eliminates the fist ball, but does not itself intesect B. Recall the definition of the sets A 1, A 2 and A 3 in 2.2-2.4 accodingly. As befoe, on the event B = { Φ A 1, } = 1, we can define a andom vecto X, R, W coesponding to the location of the cente, the adius and the weight of the single ball in Φ fo which the pai X, R is in the set A 1. Theefoe, P B D th = P B ˆB, whee now ˆB = { W > max{w : x, t, w Φ A 2 A 3, and thee is x, t, w Φ A 1 A 2 A 3 c and w > W such that B t x intesects B R X }. A standad computation shows that if fo example 2 R 3, then the expected numbe of the Boolean balls that intesect B R X but not B is at least c d. Similaly, P Φ A 1, = 1, 2 R 3 c d F. Since m A 2 A 3, c d, in ode to pove the lowe bound in 2.21 it is enough to pove the following statement. Let c 1, c 2 be positive numbes, and let N 1, N 2 be independent Poisson andom vaiables with means c 1 d and c 2 d, coespondingly. Let W, W n 1, n = 1, 2,..., W n 2, n = 1, 2,... be i.i.d. standad unifom andom vaiables independent of the Poisson andom vaiables. Then fo some positive c, 2.22 P sup W n 2 > W > sup W n 1 c d. n N 2 n N 1

CONTACT DISTRIBUTION IN A THINNED MODEL 11 To this end, note that, by symmety, fo any fixed n 1 1, n 2 1, 2.23 P sup W n 2 > W > sup W n 1 = n n 2 n n 1 Now 2.22 follows fom the fact that n 2 n 1 + n 2 + 1 P N 2 c 2 /2 d 1, P N 1 2c 1 d 1 1 n 1 + 1. as. This completes the poof of the lowe bound. Now we pove the uppe bound in 2.21. Let K 1 be a lage positive numbe we will specify below. Denote A 7 K 1 = { x, s, w R d,, : s /K 1, the ball B s x intesects the ball B }. As in 2.9 we have 2.24 E [ Φ A 7 K 1 ] c d F with a K 1 -dependent constant c. Theefoe, if l 1 α/α d, then P Φ A 7 K 1 > l 1 = o F as, and, hence, we need to pove that 2.25 lim sup Fix θ d/α, 1, and let As above, we have fo lage, P B D th =, Φ A 7 K 1 l 1 <. F A 8 K 1, θ = { x, s, w R d,, : θ < s < /K 1, the ball B s x intesects the ball B }. E [ Φ A 8 K 1, θ ] c d F θ. By the choice of θ we see that, if l 2 α/θα d, then P Φ A 8 K 1, θ > l 2 = o F as. Theefoe, in ode to establish 2.25, it is enough to pove that fo evey j =, 1,..., l 1, P B D th =, Φ A 7 K 1 = j, Φ A 8 K 1, θ l 2 2.26 lim sup <. F Now we specify K 1 by setting K 1 > l 2. Note that fo evey choice of l 2 and K 1 as above, the complement in B of the union of at most l 2 balls of adii not exceeding /K 1 contains a ball of a adius γ fo some γ = γl 2, K 1 > we can choose γ = 1/l 2 1/K 1 /2. We fist conside the case j = in 2.26. Let Φ,θ be the estiction of the Poisson andom measue Φ to the set R d, θ ],. Using, once again, the popety of a Poisson measue that its estictions to disjoint sets ae independent, we see that 2.26 with j = will follow once we show that P B γ D,θ,th = 2.27 lim sup <, F whee D,θ,th is the gain cove coesponding to the thinning of Φ,θ with the weights still being independent of the adii. The fact that we ae allowed to use the ball centeed at the oigin in

12 YINGHUA DONG AND GENNADY SAMORODNITSKY 2.27, instead of a andomly centeed ball descibed in the pevious paagaph, is a consequence of tanslation invaiance of Φ,θ. In ode to pove 2.27, we need one moe simple estimate. Let t be a lage numbe, t γ/3. Conside concentic balls B t/3, B t and B 3t. Then thee is < q < 1 such that 2.28 P sup{w : x, s, w Φ,θ Bt \ B t/3, } > max sup{w : x, s, w Φ,θ Bt/3 \ B t/9, }, sup{w : x, s, w Φ,θ B3t \ B t, } q fo all t lage enough. Indeed, the Poisson andom measue Φ,θ assigns mean measues of the ode ct d to each of the thee annuli in question with c K 1 -dependent, so 2.28 follows by using conditioning and a computation analogous to 2.23. Now it is clea that the pobability in the numeato in 2.27 can be bounded fom above by 1 q c1 θ fo some c > because we can fit into B γ tiple annuli as above with the adial sepaation between neighboing tiples exceeding θ, which makes, by the definition of Φ,θ, the events whose pobabilities ae computed in 2.28, independent. Theefoe, 2.27 holds, and so we have poved 2.26 with j =. Next we conside 2.26 with j = 1. It follows fom 2.24 that P Φ A 7 K 1 = 1 = O d F as. Theefoe, we need to pove the following vesion of 2.27: conside the gain cove D,θ,th and a andom vaiable W independent of it, whose law is the distibution of the weight in the Boolean model W is the weight of the single ball in Φ A 7 K 1. We eliminate all the balls in D,θ,th whose weight is smalle o equal to W, and we call the esulting gain cove ˆD,θ,th. Then 2.26 with j = 1 will follow once we pove that 2.29 lim sup d P B γ ˆD,θ,th = <. In ode to see that this is tue, we use an agument simila to the one used to pove 2.27. Conside the tiple annuli in 2.28. Since we aleady know that the pobability that fewe than c 1 θ events in 2.28 occus is o d, we only need to conside what happens if at least c 1 θ of the events occu. In the latte case, the only possibility fo B γ ˆD,θ,th = is that the weight of the heaviest Φ-ball in the union of c 1 θ of annuli of adii of ode c and width of ode c θ does not exceed W. Since the mean measue of the Poisson andom measue Φ,θ assigns the weight of the ode c d to that union, the latte pobability does not exceed c d, once again, by conditioning and a computation analogous to 2.23. Theefoe, 2.29 is tue, and so we have poved 2.26 with j = 1. The cases j = 2,..., l 1, ae simila and easie, since the pobabilities P Φ A 7 K 1 = j become asymptotically smalle as j inceases. This completes the poof of the uppe bound in 2.21. 3. Heavy small balls and exponential decay of the contact distibution In this section we pove that, if the weight of a ball is a stictly deceasing function of its adius, then the tail of the contact distibution decays exponentially fast. This tuns out to be unelated to the fact that the tail of the adii of the balls is egulaly vaying.

CONTACT DISTRIBUTION IN A THINNED MODEL 13 Theoem 3.1. Assume that the distibution of the adii of the balls in the Boolean model satisfies 1.2. If lage balls have smalle weights, then fo some c > the contact distibution of the thinned gem-gain model satisfies 3.1 H e c d fo all lage enough. Poof. Once again, we wok with the tail of the empty space function. Since the weight of a ball is a function of its adius, we switch, once again, to viewing the Poisson andom measue Φ as a measue in the d + 1-dimensional space R d,. Choose a finite numbe γ > such that F, γ >. Let Φ γ be the estiction of the Poisson andom measue Φ to the set {x, s : s γ}. As in the poof of Theoem 2.2, fo some c > that depends on γ, we can find at least c d disjoint balls of adius γ within B, such that the distance between any two diffeent balls exceeds 2γ. Let us call these balls B i, i = 1,..., n, with n c d. Fo i = 1,..., n conside the event { H i = {Φ γ x, s : Bs x B i } { = 1, Φ γ x, s : Bs x [ B i + B γ ] } } \ B i =. Hee B i + B γ is simply the ball concentic with B i of adius 2γ. Note that, on the event H i, the single Φ γ -ball in the desciption on the event cannot be eliminated by any othe Φ-ball. Indeed, if anothe Φ γ -ball intesected it, the latte ball would be in the set [ B i + B γ ] \ B i, which is impossible on the event H i. Futhemoe, any Φ-ball which is not a Φ γ -ball has simply too lage a adius. Theefoe, P B D th = P n i=1hi c = 1 P H 1 n 1 P H1 c d. The equality in this calculation follows fom the fact that the balls B i ae sufficiently fa away fom each othe so that diffeent events H i ae detemined by estictions of the Poisson andom measue Φ γ to disjoint sets and, hence, ae independent. In ode to pove the theoem we only need to check that P H 1 >. Since Φ γ is tanslation invaiant, we eplace, in the calculation below, B 1 by B γ and B 1 + B γ by B 2γ. Note that the event H 1 is defined as the intesection of two independent events, so we only need to check that each one of these events has a positive pobability. It is clea that so that and, hence, Futhe, so that Φ γ Bγ, γ] Φ γ { x, s : Bs x B γ } Φ γ B2γ, γ], E [ Φ γ { x, s : Bs x B γ }], { P Φ γ x, s : Bs x B γ } = 1 >. Φ γ { x, s : Bs x B 2γ \ B γ } Φ γ B3γ, γ], E [ Φ γ { x, s : Bs x B 2γ \ B γ }] <

14 YINGHUA DONG AND GENNADY SAMORODNITSKY and, hence, { P Φ γ x, s : Bs x B 2γ \ B γ } = >. This implies that P H 1 >, and the poof of the theoem is complete. Notice that a lowe bound of the type H e cd fo lage with, possibly, a diffeent exponent c than in 3.1 is tivially tue since it holds fo the oiginal spheical Boolean model even befoe thinning. Refeences J. Andeson, O. Häggstöm and M. Mansson 26: The volume faction of a nonovelapping gem-gain model. Electonic Communications in Pobability 11:78 99. D. Hug, G. Last and W. Weil 22: A suvey on contact distibution. In Mophology of Condensed Matte, K. Mecke and D. Stoyan, editos, volume 6 of Lectue Notes in Physics. Spinge-Velag, Belin, pp. 317 357. M. Kuonen and L. Leskelä 213: Had-coe thinning of gem-gain models with powe-law gain sizes. Advances in Applied Pobability 45:595 625. M. Mansson and M. Rudemo 22: Random pattens of nonovelapping convex gains. Advances in Applied Pobability 34:718 738. T. Nguyen and F. Baccelli 213: On the geneating functionals of a class of andom packing point pocesses. Pepint. S. Resnick 27: Heavy-Tail Phenomena: Pobabilistic and Statistical Modeling. Spinge, New Yok. D. Stoyan, W. Kendall and J. Mecke 1995: Stochastic Geomety and its Applications. Wiley. College of Mathematics and Statistics, Nanjing Univesity of Infomation Science and Technology, Nanjing 2144, China E-mail addess: dongyinghua1@163.com School of Opeations Reseach and Infomation Engineeing, and Depatment of Statistical Science, Conell Univesity, Ithaca, NY 14853 E-mail addess: gs18@conell.edu