Concentration inequalities for functionals of Poisson cylinder processes
Anastas Baci, Carina Betken, Anna Gusakova, Christoph Thaele
CConcentration inequalities forfunctionals of Poisson cylinder processes
Anastas Baci ∗ , Carina Betken † , Anna Gusakova ‡ and Christoph Thäle § Abstract
Random union sets Z associated with stationary Poisson processes of k -cylinders in R d are con-sidered. Under general conditions on the typical cylinder base a concentration inequality for thevolume of Z restricted to a compact window is derived. Assuming convexity of the typical cylinderbase and isotropy of Z a concentration inequality for intrinsic volumes of arbitrary order is estab-lished. A number of special cases are discussed, for example the case when the cylinder bases arisefrom a random rotation of a fixed convex body. Also the situation of expanding windows is stud-ied. Special attention is payed to the case k = 0 , which corresponds to the classical Boolean model. Keywords . Boolean model, concentration inequality, cylindrical integral geometry, intrinsic volume,Poisson cylinder process, stochastic geometry.
MSC . Primary 60D05, 60F10; Secondary 52A22, 60E15.
The stationary Boolean model is one of the most versatile models considered in stochastic geometry. Itsnumerous applications range, for example, from coverage optimization in telecommunication networksto questions related to virtual material design. While mean value formulas for the intrinsic volumesof the Boolean model are rather classical (see, e.g., [16]), only recently a satisfactory description ofsecond-order properties was derived by Hug, Last and Schulte [10] together with an accompanyingcentral limit theory. In addition, building on a concentration inequality for Poisson functionals onabstract phase spaces Gieringer and Last [3] obtained concentration inequalities for a class of measuresassociated with a rather general Boolean model in an observation window. On their way they wereable to refine earlier estimates of Heinrich [4] for the volume of a stationary Boolean model in R d restricted to a compact observation window, which in turn were obtained by means of sharp boundson cumulants.The aim of the present paper is to prove concentration inequalities for the volume as well as for theintrinsic volumes associated with the union set of a stationary Poisson cylinder process in R d restrictedto a compact window. For k ∈ { , , . . . , d − } we understand by a k -cylinder the Minkowski sum of a k -dimensional linear subspace in R d and a compact set in its orthogonal complement. A Poisson processof k -cylinders (or Poisson cylinder process for short) is a Poisson process on the space of k -cylinders in R d . We refer to Section 2.2 for a formal description of the model. In this paper we consider the unionset Z associated with such a Poisson cylinder process, which is observed in a compact window W ⊂ R d .It is assumed throughout that Z is a stationary random closed set. In this case the distribution of Z is determined by an intensity parameter γ ∈ (0 , ∞ ) as well as the distribution Q of the pair (Ξ , Θ) ,where Ξ describes the base and Θ the direction of the typical cylinder. It is worth pointing out that theconcept of a Poisson cylinder process generalizes that of the Boolean model discussed above, which isincluded as the special case k = 0 . In this situation Ξ is the typical grain of the Boolean model and the ∗ Ruhr University Bochum, Germany. Email: [email protected] † Ruhr University Bochum, Germany. Email: [email protected] ‡ Ruhr University Bochum, Germany. Email: [email protected] § Ruhr University Bochum, Germany. Email: [email protected] a r X i v : . [ m a t h . P R ] A ug igure 1.1: Left panel: Simulation of an isotropic Poisson cylinder process in R with spherical cylinderbase. Right panel: Simulation of an anisotropic Poisson cylinder process in R with rectangular cylinderbase. Both simulations were provided by Claudia Redenbach, Kaiserslautern.random direction Θ has no relevance. Poisson cylinder processes were formally introduced by Matheron[12], Miles [13] and Weil [19]. More recently, central limit theorems for stationary Poisson processes ofcylinders were studied by Heinrich and Spiess [5, 6]. Under an exponential moment assumption on the ( d − k ) -volume of the typical cylinder base they obtained in [5] a central limit theorem for the volumeof Z in a sequence of growing windows, that is, for Z ∩ W r , where W r = rW , as r → ∞ . More precisely,using sharp bounds on cumulants they were able to deduce a rate of convergence as well as Cramér-type large deviations. In a subsequent paper [6] they were able to relax the moment assumptions andto add a central limit theorem for the surface content. Characteristic quantities like volume fraction,covariance function and contact distribution functions of anisotropic Poisson cylinder processes wereinvestigated by Spiess and Spodarev [17]. In addition, percolation and connectivity properties relatedto Poisson cylinder processes with spherical bases and k = 1 were studied by Tykesson and Windisch[18], Hilario, Sidoravicius and Teixeira [7] as well as Borman and Tykesson [1].The aim of the present paper is to derive tail bounds for the volume as well as for the intrinsic volumesof the random union set Z associated with a stationary Poisson cylinder process restricted to a compactobservation window. More precisely, under rather general assumptions on the distribution of the typicalcylinder base we derive bounds for the upper and lower tail of the volume ( d -dimensional Lebesguemeasure) F := λ d ( Z ∩ W ) of Z ∩ W , where W ⊂ R d is a compact set with positive volume. Ourbounds generalize in a natural way the results from [3] for the Boolean model. A number of specialcases are discussed separately. For example, we consider the case where the cylinder bases are randomrotations of a fixed convex body. We will see that in this situation our tail bounds are of the form P ( F − E F ≥ r ) ≤ exp( − Θ ( r log r )) , r ≥ , (1.1) P ( F − E F ≤ − r ) ≤ exp( − Θ ( r )) , ≤ r ≤ E F, (1.2)where Θ ( r log r ) stands for a quantity from O ( r log r ) ∩ Ω ( r log r ) in the usual Landau notation. Asfor the Boolean model this constitutes a significant improvement compared to the bounds that can bededuced by means of the general limit theorems for large deviations [15] from the cumulant estimatesprovided in [5].Beside the volume of Z ∩ W we also study the intrinsic volumes of Z ∩ W under the assumption that thecylinder bases are convex and that the union set Z is a stationary and isotropic random closed set. Weemphasize that the intrinsic volumes are of particular importance since every continuous, additive andmotion-invariant functional on the class of convex bodies can be represented as a linear combinationof intrinsic volumes (this is the content of Hadwiger’s theorem). We remark that compared to the2olume case the intrinsic volumes are more difficult to handle. This partially relies on the fact thateven mean value formulas for intrinsic volumes of (stationary and isotropic) Poisson cylinder processesare not available in the existing literature and needed to be developed in the present paper as well. Inaddition, for the case of intrinsic volumes, isoperimetric inequalities have to be used in order to bringthe bounds in a convenient form. As for the volume we consider especially the case where the cylinderbases are random rotations of a fixed convex body and deduce bounds which are comparable to (1.1)and (1.2). We remark that our results for intrinsic volumes are new even for the special case of theBoolean model for which only the case of the surface content was previously studied in [2] under quiterestrictive assumptions on the typical cylinder base.The remaining parts of this paper are structured as follows. In Section 2.1 we gather some notationand in Section 2.2 we recall the formal definition and description of a Poisson cylinder process and itsassociated union set. In particular, we derive there a necessary and sufficient criterion under which theunion set is isotropic. A concentration inequality for general Poisson functionals from [3] is presentedin Section 2.3. Tail bounds for the volume are the content of Section 3 and a number of special casesare discussed in Section 4. We present concentration properties for the class of intrinsic volumes in thefinal Section 5. For d ∈ N we let λ d be the Lebesgue measure on R d . The s -dimensional Hausdorff measure is denotedby H s , s ≥ . A centred Euclidean ball in R d with radius r > is denoted by B dr . The volume of the d -dimensional unit ball is given by κ d := λ d ( B d ) = π d/ Γ(1+ d/ . We let C (cid:48) ( R d ) be the space of non-emptycompact subsets of R d and recall that by a convex body K ⊂ R d we understand a compact convex setwith non-empty interior. For k ∈ { , , . . . , d } and a convex body K ⊂ R d we let V j ( K ) be the j thintrinsic volume of K . In particular, V d ( K ) = λ d ( K ) , V d − ( K ) = H d − ( ∂K ) and V ( K ) is a constantmultiple of the mean width of K . We use the symbol diam( A ) to indicate the diameter of a set A ⊂ R d .For a (possibly lower-dimensional) convex set K ⊂ R d we denote by K ∗ = − K the reflection of K at the origin. Moreover, the linear hull of A ⊂ R d is denoted by lin( A ) . By P d − k : R d → R d − k wedenote the orthogonal projection of R d to R d − k , i.e., the projection to the first d − k coordinates. By O d and SO d we denote the group of orthogonal d × d matrices and of orthogonal d × d matrices withdeterminant , respectively. Let d ≥ and k ∈ { , , . . . , d − } . Further, we let G ( d, k ) be the Grassmannian of k -dimensionallinear subspaces of R d . By a k -cylinder in R d one understands the Minkowski sum of some L ∈ G ( d, k ) with a non-empty compact subset of L ⊥ , the orthogonal complement of L . We identify a subspace L ∈ G ( d, k ) with the unique element φ L of the equivalence class Φ L of orthogonal matrices φ ∈ SO d satisfying L = φE k , where E k = lin( e d − k +1 , . . . , e d ) and e , . . . , e d is the standard orthonormal basis in R d . In fact, one can choose for φ L the lexicografically smallest element of the compact set Φ L , whichyields a one-to-one correspondence between G ( d, k ) and SO d,k := { φ L = lex min Φ L : L ∈ G ( d, k ) } upto orientation of the subspaces, cf. [5, 6]. In particular, this allows us to regard SO d,k as a compacthomogeneous space for SO d .Fix γ ∈ (0 , ∞ ) and let η be a stationary Poisson process on lin( e , . . . , e d − k ) ⊂ R d with intensity γ .Let C (cid:48) d − k be the space of non-empty compact subsets of lin( e , . . . , e d − k ) . Here and in what follows, weidentify lin( e , . . . , e d − k ) with R d − k ⊂ R d . We define M d,k := SO d,k × C (cid:48) d − k and let Q be a probabilitymeasure on M d,k . By ξ we denote an independent Q -marking of η , which is a Poisson process onthe product space R d − k × M d,k with intensity measure γ λ d − k ⊗ Q , cf. [11]. Further we denote by (Θ , Ξ) ∈ M d,k a random pair with distribution Q . It represents the (not necessarily independent)distribution of the direction and the base of the typical cylinder in the usual sense of Palm theory.3y a stationary Poisson process of k -cylinders with intensity γ and base-direction distribution Q weunderstand the point process (cid:101) ξ := (cid:88) ( x,θ,K ) ∈ ξ δ Z ( x,θ,K ) , Z ( x, θ, K ) = θ (( K + x ) × E k ) on the space of k -cylinders in R d , where δ ( · ) denotes the Dirac measure, cf. [5, 6]. In this paper weare interested in the random union set Z := (cid:91) X ∈ (cid:101) ξ X = (cid:91) ( x,θ,K ) ∈ ξ Z ( x, θ, K ) induced by the stationary marked Poisson process ξ or the stationary Poisson cylinder process (cid:101) ξ ,respectively, where we write X ∈ (cid:101) ξ to indicate that X belongs to the support of (cid:101) ξ . It is known from[5, 6] that Z is a random closed subset of R d in the usual sense of stochastic geometry [16, Chapter 2],provided that E λ d − k (Ξ + B d − kε ) < ∞ for some ε > . (2.1)In this case, F := λ d ( Z ∩ W ) is a well-defined random variable for any compact subset W ⊂ R d . Inwhat follows we shall assume that (2.1) is always satisfied.Another point we shall discuss here is the isotropy property of the random union set Z , which meansthat ρZ has the same distribution as Z for all ρ ∈ SO d . While for the Boolean model an isotropycriterium is well known, surprisingly we were not able to locate a necessary and sufficient condition forisotropy of Z in the existing literature. Lemma 2.1.
The random closed set Z is isotropic if and only if Q ( SO d,k × · ) is an O d − k -invariantprobability measure on C (cid:48) d − k and Q ( · × C (cid:48) d − k ) is the SO d -invariant Haar probability measure on SO d,k .Proof. We recall that the capacity functional T X of a random closed set X is given by T X ( C ) := P ( X ∩ C (cid:54) = ∅ ) , C ∈ C (cid:48) ( R d ) . According to [16, Theorem 2.4.5] a random closed set X is isotropic if andonly if its capacity functional is rotation invariant, that is, if T X ( C ) = T X ( ρC ) holds for all ρ ∈ SO d and C ∈ C (cid:48) ( R d ) . The capacity functional T Z of Z is known and given by T Z ( C ) = 1 − exp (cid:0) − γ E λ d − k ( P d − k (Θ T C ) + Ξ ∗ ) (cid:1) according to [17, Lemma 1] or the results in [5, Section 5]. Now, for ρ ∈ SO d consider − T Z ( ρC ) = exp (cid:0) − γ E λ d − k ( P d − k (Θ T ( ρC )) + Ξ ∗ ) (cid:1) and note that E λ d − k (cid:0) P d − k (Θ T ( ρC )) + Ξ ∗ (cid:1) = (cid:90) M d,k λ d − k (cid:0) P d − k ( θ T ( ρC )) + K ∗ (cid:1) Q ( d ( θ, K ))= (cid:90) M d,k λ d − k (cid:0) P d − k (( ρ T θ ) T ( C )) + K ∗ (cid:1) Q ( d ( θ, K )) . (2.2)It was mentioned in [5] that the space SO d,k is the same as the space of representatives of the quotientspace SO d / S ( O d − k × O k ) , where S ( O d − k × O k ) can be identified with the following space of blockmatrices: S ( O d − k × O k ) = (cid:26)(cid:18) A B (cid:19) : A ∈ O d − k , B ∈ O k , det A = det B (cid:27) . By construction of SO d,k as a space of canonical representatives, this means that every element ρ ∈ SO d admits a unique decomposition ρ T = ρ d,k ρ d − k ρ k (2.3)4here ρ d,k ∈ SO d,k , ρ d − k ∈ (cid:101) O d − k and ρ k ∈ (cid:101) O k . Here, (cid:101) O d − k and (cid:101) O k are the sets of block matrices givenby (cid:101) O d − k := (cid:26)(cid:18) A I k (cid:19) : A ∈ O d − k (cid:27) , (cid:101) O k := (cid:26)(cid:18) I d − k B (cid:19) : B ∈ O k (cid:27) with I n being n × n identity matrix, n ∈ N . For any θ ∈ SO d,k there are uniquely determined elements ρ θd,k ∈ SO d,k , ρ θd − k ∈ (cid:101) O d − k and ρ θk ∈ (cid:101) O k such that ρ T θ = ρ θd,k ρ θd − k ρ θk . (2.4)Plugging this into (2.2) yields E λ d − k (cid:0) P d − k (Θ T ( ρC )) + Ξ ∗ (cid:1) = (cid:90) M d,k λ d − k (cid:16) P d − k (( ρ θd,k ρ θd − k ρ θk ) T ( C )) + K ∗ (cid:17) Q ( d ( θ, K ))= (cid:90) M d,k λ d − k (cid:16) P d − k (( ρ θk ) T ( ρ θd − k ) T ( ρ θd,k ) T ( C )) + K ∗ (cid:17) Q ( d ( θ, K ))= (cid:90) M d,k λ d − k (cid:16) P d − k (( ρ θd,k ) T ( C )) + ( ρ θd − k )( K ∗ ) (cid:17) Q ( d ( θ, K ))= (cid:90) M d,k λ d − k (cid:16) P d − k (( ρ θd,k ) T ( C )) + K ∗ (cid:17) Q ( d ( θ, K )) . Here, to obtain the third equality we used the fact that ( ρ θk ) T does not influence the projection P d − k ( · ) ,that ρ θd − k acts in R d − k and thus commutes with the projection P d − k , and that the Lebesgue measure λ d − k is O d − k -invariant. Moreover, the last equality follows from the fact that Q ( SO d,k × · ) is O d − k -invariant by assumption.To simplify the last expression further, we apply twice the change-of-basis formula from linear algebra.This implies the relations ρ θd − k = θ T ρ d − k θ and ρ θk = θ T ρ k θ. Putting this together with (2.3) and (2.4) we conclude that ρ d,k ρ d − k ρ k θ = ρ T θ = ρ θd,k ( θ T ρ d − k θ )( θ T ρ k θ ) = ρ θd,k θ T ρ d − k ρ k θ, and hence ρ θd,k = ρ d,k θ . From this we obtain − T Z ( ρC ) = exp (cid:16) − γ (cid:90) M d,k λ d − k (cid:0) P d − k (( ρ d,k θ ) T ( C )) + K ∗ (cid:1) Q ( d ( θ, K )) (cid:17) = exp (cid:16) − γ (cid:90) M d,k λ d − k (cid:0) P d − k ( θ T ( C )) + K ∗ (cid:1) Q ( d ( θ, K )) (cid:17) = 1 − T Z ( C ) , where we have used our assumption that Q ( · × C (cid:48) d − k ) is the SO d -invariant Haar probability measureon SO d,k . This concludes the proof.For a measurable set M ⊆ M d,k and ρ ∈ SO d we define ρM := { ( ρ θd,k , ρ θd − k K ) : ( θ, K ) ∈ M } , where we applied the decomposition ρθ = ρ θd,k ρ θd − k ρ θk with ρ θd,k ∈ SO d,k , ρ θd − k ∈ (cid:101) O d − k and ρ θk ∈ (cid:101) O k using the notation introduced in the previous proof. We say that a probability measure Q on M d,k is5otation invariant, provided that Q ( ρM ) = Q ( M ) for all measurable M ⊆ M d,k . Repeating the sameargument as in the proof of Lemma 2.1 we can conclude that Q is rotation invariant if and only if Q ( SO d,k × · ) is an O d − k -invariant probability measure on C (cid:48) d − k and Q ( · × C (cid:48) d − k ) is the SO d -invariantHaar probability measure on SO d,k . Especially, from Lemma 2.1 we conclude that the random unionset Z is isotropic if and only if Q is rotation invariant. In this section we rephrase a general concentration inequality for Poisson functionals that was recentlyproved in [3] using a covariance identity for functionals of Poisson processes on abstract phase spacesand a classical Chernoff-type argument. For this we let ( X , X ) be a measurable space and Λ be some σ -finite measure on X . By η we denote a Poisson process on X with intensity measure Λ , which isdefined over some probability space (Ω , A , P ) , cf. [11]. By N = N ( X ) we denote the space of σ -finitecounting measures on X , which is supplied with the σ -field N induced by the vague topology on N .We denote the distribution on N of a Poisson process with intensity measure Λ by Π Λ . Finally, by aPoisson functional we understand a random variable F P -almost surely satisfying F = f ( η ) for somemeasurable function f : N → R , called a representative of F .For a measurable function f : N → R and a point x ∈ X we define the first-order difference (oradd-one-cost) operator by D x f ( µ ) = f ( µ + δ x ) − f ( µ ) , µ ∈ N . In particular, for a Poisson functional F with representative f we write D x F for D x f ( η ) . For asquare-integrable Poisson functional F ∈ L ( P ) we define s F := sup { s ≥ e sF ∈ L ( P ) , De sF ∈ L ( P ⊗ Λ) } ∈ [0 , ∞ ] , (2.5)and for s ∈ [0 , s F ) we put V F ( s ) := (cid:90) X ( e sD x F − (cid:90) (cid:90) N D x f ( η t + µ ) Π (1 − t )Λ ( d µ ) d t Λ( d x ) , (2.6)where η t , t ∈ [0 , denotes a t -thinning of η (which is a Poisson process on X with intensity measure t Λ ). We are now in the position to rephrase the concentration inequality from [3, Corollary 2.3]. Lemma 2.2.
Let F = f ( η ) ∈ L ( P ) be a Poisson functional such that DF ∈ L ( P ⊗ Λ) . Assume that P -almost surely V F ( s ) ≤ v ( s ) for some measurable function v : [0 , s F ) → R . Then P ( F − E F ≥ r ) ≤ exp (cid:16) inf s ∈ [0 ,s F ) (cid:16) s (cid:90) v ( u ) d u − rs (cid:17)(cid:17) , r ≥ . As already remarked in [3] a similar inequality holds for the lower tail of the distribution of F if s F and V F ( s ) from (2.5) and (2.6) are replaced by s (lt) F := s − F and V (lt) F ( s ) := V − F ( s ) , respectively. Inparticular, note that for s ∈ [0 , s (lt) F ) the identity V (lt) F ( s ) = (cid:90) X (1 − e − sD x F ) (cid:90) (cid:90) N D x f ( η t + µ ) Π (1 − t )Λ ( d µ ) d t Λ( d x ) holds. Lemma 2.3.
Let F = f ( η ) ∈ L ( P ) be such that DF ∈ L ( P ⊗ Λ) . Assume that P -almost surely V (lt) F ( s ) ≤ v ( s ) for some measurable function v : [0 , s (lt) F ) → R . Then P ( F − E F ≤ − r ) ≤ exp (cid:16) inf s ∈ [0 ,s (lt) F ) (cid:16) s (cid:90) v ( u ) d u − rs (cid:17)(cid:17) , r ≥ . A concentration inequality for the volume
Our goal in this section is to apply the concentration inequalities for general Poisson functionals fromSection 2.3 to the volume of the union set of a stationary Poisson cylinder process within a boundedwindow. More precisely, we let Z be the union set of a stationary Poisson process of k -cylinders in R d with intensity γ ∈ (0 , ∞ ) and base-direction distribution Q . We assume that all random quantitiesconsidered are defined over some probability space (Ω , A , P ) . Moreover, we let W ⊂ R d be a compactset with λ d ( W ) > . We are interested in the Poisson functional F := λ d ( Z ∩ W ) , i.e., F is the total volume ( d -dimensional Lebesgue measure) of all cylinders within W . In this sectionand the next section we assume that the volume of the typical cylinder base has positive and finitefirst moment, i.e., m d − k := E λ d − k (Ξ) ∈ (0 , ∞ ) . In order to check the assumptions of Lemma 2.2, an analysis of the first-order difference operator of F is necessary. We start by observing that the additivity of the Lebesgue measure implies that, for ( λ d − k ⊗ Q ) -almost all ( x, θ, K ) ∈ R d − k × M d,k , the following equality D ( x,θ,K ) F = λ d (( Z ∪ Z ( x, θ, K )) ∩ W ) − λ d ( Z ∩ W )= λ d ( Z ∩ W ) + λ d ( Z ( x, θ, K ) ∩ W ) − λ d ( Z ∩ Z ( x, θ, K ) ∩ W ) − λ d ( Z ∩ W )= λ d ( Z ( x, θ, K ) ∩ W ) − λ d ( Z ∩ Z ( x, θ, K ) ∩ W ) holds P -almost surely. Using this representation for the difference operator, we can prove the followingtechnical result, where we recall the definition of the quantity s F from (2.5) and also that s (lt) F = s − F . Lemma 3.1.
Under the assumptions mentioned above we have that F ∈ L ( P ) , DF ∈ L ( P ⊗ λ d − k ⊗ Q ) , s F = ∞ and s (lt) F = ∞ .Proof. Corollary 18.8 in [11] shows that E ( F ) ≤ ( E ( F )) + γ (cid:90) R d − k (cid:90) M d,k E [( D ( x,θ,K ) F ) ] Q ( d ( θ, K )) λ d − k ( d x ) ≤ λ d ( W ) + γ (cid:90) R d − k (cid:90) M d,k λ d ( Z ( x, θ, K ) ∩ W ) Q ( d ( θ, K )) λ d − k ( d x ) ≤ λ d ( W ) + γ λ d ( W ) ( λ d − k ⊗ Q ) (cid:16) { ( x, θ, K ) ∈ R d − k × M d,k : Z ( x, θ, K ) ∩ W (cid:54) = ∅ } (cid:17) < ∞ , since W is compact and λ d − k ⊗ Q is a locally finite measure on R d − k × M d,k . This shows that F ∈ L ( P ) and at the same time DF ∈ L ( P ⊗ λ d − k ⊗ Q ) .Next, we let s ≥ and observe that P -almost surely and for ( λ d − k ⊗ Q ) -almost all ( x, θ, K ) ∈ R d − k × M d,k , D ( x,θ,K ) e sF = e sλ d (( Z ∪ Z ( x,θ,K )) ∩ W ) − e sλ d ( Z ∩ W ) = e sF (cid:0) e sD ( x,θ,K ) F − (cid:1) . E (cid:90) R d − k (cid:90) M d,k (cid:0) D ( x,θ,K ) e sF (cid:1) Q ( d ( θ, K )) λ d − k ( d x )= (cid:90) R d − k (cid:90) M d,k E (cid:2)(cid:0) e sF ( e sD ( x,θ,K ) F − (cid:1) (cid:3) Q ( d ( θ, K )) λ d − k ( d x ) ≤ E [ e sF ] (cid:90) R d − k (cid:90) M d,k (cid:0) e sλ d ( Z ( x,θ,K ) ∩ W ) − (cid:1) Q ( d ( θ, K )) λ d − k ( d x ) ≤ e sλ d ( W ) (cid:0) e sλ d ( W ) − (cid:1) ( λ d − k ⊗ Q ) (cid:16) { ( x, θ, K ) ∈ R d − k × M d,k : Z ( x, θ, K ) ∩ W (cid:54) = ∅ } (cid:17) . Since the last expression is finite for all s ≥ , we conclude that De sF ∈ L ( P ⊗ λ d − k ⊗ Q ) . Moreover,using that E (cid:90) R d − k (cid:90) M d,k e sF Q ( d ( θ, K )) λ d − k ( d x ) ≤ e sλ d ( W ) ( λ d − k ⊗ Q ) (cid:16) { ( x, θ, K ) ∈ R d − k × M d,k : Z ( x, θ, K ) ∩ W (cid:54) = ∅ } (cid:17) , we obtain the assertion that s F = ∞ . To prove that s (lt) F = ∞ , we first observe that, since P -almostsurely e − sF ≤ , necessarily e − sF ∈ L ( P ) for any s ≥ . In addition, similarly to the above argument,we have that E (cid:90) R d − k (cid:90) M d,k (cid:0) D ( x,θ,K ) e − sF (cid:1) Q ( d ( θ, K )) λ d − k ( d x ) ≤ ( λ d − k ⊗ Q ) (cid:16) { ( x, θ, K ) ∈ R d − k × M d,k : Z ( x, θ, K ) ∩ W (cid:54) = ∅ } (cid:17) < ∞ for any s ≥ . This shows that s (lt) F = ∞ .Recall that E F = λ d ( W )(1 − e − γ m d − k ) =: λ d ( W ) p, (3.1)where p = 1 − e − γ m d − k is the volume fraction of the random union set Z , see [5, 17]. The main resultof this section is the following bound for the upper and the lower tail of the volume of the union set ofa stationary Poisson cylinder process in a window W . Theorem 3.2.
For all r ≥ , one has that P ( F − E F ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d − k E (cid:104) λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k ) (cid:105) − rs (cid:17)(cid:17) , and for ≤ r ≤ E F one has that P ( F − E F ≤ − r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d − k E (cid:104) λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( − sλ d − k (Ξ) diam( W ) k ) (cid:105) − rs (cid:17)(cid:17) , where Ψ( x ) = e x − x − , x ∈ R . Remark 3.3. (i) Note that the condition ≤ r ≤ E F in the inequality for the lower tail is notstrictly necessary. However, the inequality becomes trivial for all r > E F , since F can only takenon-negative values. 8ii) Taking k = 0 , Z is nothing else than a Boolean model based on a stationary Poisson pointprocess on R d with intensity γ and typical grain Ξ . In this case the two inequalities in Theorem3.2 reduce to P ( F − E F ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d E [ λ d ( W + Ξ ∗ )Ψ( sλ d (Ξ))] − rs (cid:17)(cid:17) , r ≥ , (3.2)and P ( F − E F ≤ − r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d E [ λ d ( W + Ξ ∗ )Ψ( − sλ d (Ξ))] − rs (cid:17)(cid:17) , ≤ r ≤ E F. In this form they are known from [2, 3] and our result can be seen as a natural generalization togeneral k ∈ { , . . . , d − } .(iii) Consider the degenerate case where P -almost surely Ξ = { } , which is not covered by Theorem3.2. Then Z is the union set associated with a stationary Poisson process of k -flats in R d , see [16,Section 4.4]. For simplicity assume that Z is isotropic. In this case one can consider for compactand convex W ⊂ R d with λ d ( W ) > the k -dimensional Hausdorff measure F = H k ( Z ∩ W ) .The difference operator is then given by H k ( W ∩ θ ( x + E k )) , independently of Z . One can thusapply the general concentration inequality [21, Proposition 3.1] in combination with Crofton’sformula [16, Theorem 5.1.1] from integral geometry to conclude that P ( F − E F ≥ r ) ≤ exp (cid:16) − r b log (cid:16) bra (cid:17)(cid:17) , r ≥ with b = (cid:0) diam( W )2 (cid:1) k κ k and a = γ κ k κ d − k ( dk ) κ d (cid:0) diam( W )2 (cid:1) k V d − k ( W ) . With different constants a and b this can also be established along the lines of the proof of Theorem 3.2. Proof of Theorem 3.2 – Upper tail.
In the firs step, we deduce an upper bound for the function V F ( s ) , s ≥ , defined in (2.6). We start by putting Λ := γ λ d − k ⊗ Q and considering the term T t := (cid:90) N D ( x,θ,K ) f ( η t + µ ) Π (1 − t )Λ ( d µ ) , t ∈ [0 , , where we denote by f a representative of F . By the superposition property of Poisson processes, theinequality D ( x,θ,K ) f ( η t + µ ) ≤ D ( x,θ,K ) f ( µ ) holds for ( λ d − k ⊗ Q ) -almost all ( x, θ, K ) and all t ∈ [0 , . Thus, we have that T t ≤ λ d ( Z ( x, θ, K ) ∩ W ) − (cid:90) N λ d ( Z ( µ ) ∩ Z ( x, θ, K ) ∩ W ) Π (1 − t )Λ ( d µ )= λ d ( Z ( x, θ, K ) ∩ W ) − λ d ( Z ( x, θ, K ) ∩ W ) (cid:0) − e − (1 − t ) γ E λ d − k (Ξ) (cid:1) = λ d ( Z ( x, θ, K ) ∩ W ) e − (1 − t ) γ m d − k . As a consequence and by using Fubini’s theorem, we find that V F ( s ) ≤ γ (cid:90) R d − k (cid:90) M d,k (cid:0) e sλ d ( Z ( x,θ,K ) ∩ W ) − (cid:1) (cid:90) T t d t Q ( d ( θ, K )) λ d − k ( d x ) ≤ γ (cid:90) R d − k (cid:90) M d,k (cid:0) e sλ d ( Z ( x,θ,K ) ∩ W ) − (cid:1) × (cid:90) λ d ( Z ( x, θ, K ) ∩ W ) e − (1 − t ) γ m d − k d t Q ( d ( θ, K )) λ d − k ( d x )= pm d − k v ( s ) , v ( s ) := (cid:90) R d − k (cid:90) M d,k (cid:0) e sλ d ( Z ( x,θ,K ) ∩ W ) − (cid:1) λ d ( Z ( x, θ, K ) ∩ W ) Q ( d ( θ, K )) λ d − k ( d x ) and we recall that p = 1 − e − γ m d − k is the volume fraction of the random set Z .In a next step, we shall provide an upper bound for the integral w ( s ) := pm d − k s (cid:90) v ( u ) d u. We have that m d − k p w ( s ) = (cid:90) R d − k (cid:90) M d,k s (cid:90) (cid:0) e uλ d ( Z ( x,θ,K ) ∩ W ) − (cid:1) λ d ( Z ( x, θ, K ) ∩ W ) d u Q ( d ( θ, K )) λ d − k ( d x )= (cid:90) R d − k (cid:90) M d,k (cid:104) e sλ d ( Z ( x,θ,K ) ∩ W ) − sλ d ( Z ( x, θ, K ) ∩ W ) − (cid:105) Q ( d ( θ, K )) λ d − k ( d x ) . Since λ d ( Z ( x, θ, K ) ∩ W ) ≤ λ d − k ( K ) diam( W ) k (3.3)we obtain, using Fubini’s theorem and the fact that the function Ψ( x ) = e x − x − is increasing for x > , that m d − k p w ( s ) ≤ (cid:90) M d,k (cid:104) e sλ d − k ( K ) diam( W ) k − sλ d − k ( K ) diam( W ) k − (cid:105) × (cid:90) R d − k { Z ( x, θ, K ) ∩ W (cid:54) = ∅ } λ d − k ( d x ) Q ( d ( θ, K )) . Now, for any fixed θ ∈ SO d,k we have that (cid:90) R d − k { Z ( x, θ, K ) ∩ W (cid:54) = ∅ } λ d − k ( d x )= (cid:90) R d − k { ( K + x ) ∩ P d − k ( θ T W ) (cid:54) = ∅ } λ d − k ( d x )= λ d − k ( P d − k ( θ T W ) + K ∗ ) . (3.4)Thus, we obtain w ( s ) ≤ pm d − k (cid:90) M d,k λ d − k ( P d − k ( θ T W ) + K ∗ ) Ψ( sλ d − k ( K ) diam( W ) k ) Q ( d ( θ, K ))= pm d − k E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )] . Combining now the general concentration inequality in Lemma 2.2 with Lemma 3.1 and the aboveinequality finishes the proof for the upper tail.
Proof of Theorem 3.2 – Lower tail.
A slight adaption of the proof for the upper tail also shows thefollowing bound for V (lt) F ( s ) , which will be used to control the lower tail of the Poisson functional F .Namely, replacing F by − F we have already shown in the proof of the bound for the upper tail that V (lt) F ( s ) ≤ pm d − k v (lt) ( s ) , v (lt) ( s ) = (cid:90) R d − k (cid:90) M d,k (cid:0) − e − sλ d ( Z ( x,θ,K ) ∩ W ) (cid:1) λ d ( Z ( x, θ, K ) ∩ W ) Q ( d ( θ, K )) λ d − k ( d x ) . Now, we compute m d − k p w (lt) ( s ) := m d − k p s (cid:90) v (lt) ( u ) d u = (cid:90) R d − k (cid:90) M d,k (cid:104) e − sλ d ( Z ( x,θ,K ) ∩ W ) + sλ d ( Z ( x, θ, K ) ∩ W ) − (cid:105) Q ( d ( θ, K )) λ d − k ( d x ) . Using (3.3), Fubini’s theorem, the fact that the function Ψ( x ) is decreasing for x < , and (3.4) wefind that m d − k p w (lt) ( s ) ≤ (cid:90) M d,k (cid:104) e − sλ d − k ( K ) diam( W ) k + sλ d − k ( K ) diam( W ) k − (cid:105) × (cid:90) R d − k { Z ( x, θ, K ) ∩ W (cid:54) = ∅ } λ d − k ( d x ) Q ( d ( θ, K ))= (cid:90) M d,k λ d − k ( P d − k ( θ T W ) + K ∗ ) Ψ( − sλ d − k ( K ) diam( W ) k ) Q ( d ( θ, K ))= E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ ) Ψ( − sλ d − k (Ξ) diam( W ) k )] . Combining this with Lemma 2.3 and Lemma 3.1 we finish the proof of the lower tail.
In this section we will consider a number of special choices for the window W as well as for thedistribution of the typical cylinder base Ξ when the general estimates in Theorem 3.2 can be mademore explicit. To simplify the discussion, we will also assume that the window W and the typicalcylinder base Ξ are convex bodies ( P -almost surely). We start with the situation in which the cylinder bases are random rotations and dilatation of afixed convex body M ⊂ R d − k . More precisely, if U ∈ SO d − k is a uniform random rotation in R d − k (distributed according to the unique rotationally invariant Haar probability measure ν d − k on SO d − k )and if R is a non-negative random variable with law P R then Ξ = U ( RM ) , where we assume that U and R are independent. In addition, we assume that the direction Θ of the typical cylinder base isalso uniformly distributed on SO d,k according to the unique SO d -invariant Haar probability measure ν d,k on SO d,k , independently of U and R . We note that in this situation m d − k = λ d − k ( M ) E R d − k and p = 1 − e − γλ d − k ( M ) E R d − k . Corollary 4.1.
Let the assumptions just described prevail. Assume that E [ R d − k e sR d − k ] < ∞ for some s > , and assume that λ d − k ( M ) ∈ (0 , ∞ ) . Then P ( F − E F ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:104) pm d − k d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) E [ R d − k − j Ψ( αsR d − k )] − rs (cid:105)(cid:17) or any r ≥ , where α := λ d − k ( M ) diam( W ) k . Moreover, assuming that E R d − k ) < ∞ we have that P ( F − E F ≤ − r ) ≤ exp (cid:16) inf s ≥ (cid:104) pm d − k d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) E [ R d − k − j Ψ( − αsR d − k )] − rs (cid:105)(cid:17) for any ≤ r ≤ E [ F ] .Proof. We need to investigate the term E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )] , which shows up in the exponents in Theorem 3.2. Using Fubini’s theorem, the assumed independenceproperties of Θ and Ξ , the scaling property of the Lebesgue measure and the invariance of the Lebesguemeasure under rotations we have that E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )]= ∞ (cid:90) (cid:90) SO d − k (cid:90) SO d,k λ d − k ( P d − k ( θ T W ) + (cid:37) ( rM ∗ )) Ψ( sλ d − k ( (cid:37) ( rM )) diam( W ) k ) × ν d,k ( d θ ) ν d − k ( d (cid:37) ) P R ( d r )= ∞ (cid:90) (cid:90) SO d,k (cid:90) SO d − k λ d − k ( P d − k ( θ T W ) + (cid:37) ( rM ∗ )) Ψ( sr d − k λ d − k ( M ) diam( W ) k ) × ν d − k ( d (cid:37) ) ν d,k ( d θ ) P R ( d r ) . Since Ψ( sr d − k λ d − k ( M ) diam( W ) k ) is independent of (cid:37) , the inner integral can be evaluated by meansof the rotational integral formula from [16, Theorem 6.1.1]. This yields (cid:90) SO d − k λ d − k ( P d − k ( θ T W ) + (cid:37) ( rM ∗ )) ν d − k ( d (cid:37) )= d − k (cid:88) j =0 κ d − k − j κ j (cid:0) d − kj (cid:1) κ d − k V j ( P d − k ( θ T W )) r d − k − j V d − k − j ( M ) , where we also used the homogeneity of the intrinsic volumes. Thus, E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )]= d − k (cid:88) j =0 κ d − k − j κ j (cid:0) d − kj (cid:1) κ d − k V d − k − j ( M ) ∞ (cid:90) r d − k − j Ψ( sr d − k λ d − k ( M ) diam( W ) k ) × (cid:90) SO d,k V j ( P d − k ( θ T W )) ν d,k ( d θ ) P R ( d r ) . Using now the mean projection formula for intrinsic volumes [16, Theorem 6.2.2] we conclude from thedefinition of SO d,k and the uniqueness of Haar measures that (cid:90) SO d,k V j ( P d − k ( θ T W )) ν d,k ( d θ ) = (cid:90) G ( d,d − k ) V j ( P L ( W )) ν G ( d,d − k ) ( d L ) = (cid:0) d − jk (cid:1) κ d − j κ d − k (cid:0) dk (cid:1) κ d − k − j κ d V j ( W ) , P L ( W ) denotes the orthogonal projection of W onto L ∈ G ( d, d − k ) and ν G ( d,d − k ) stands forthe unique Haar probability measure on the Grassmannian G ( d, d − k ) . As a consequence, we concludethat E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )]= d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) ∞ (cid:90) r d − k − j Ψ( sr d − k λ d − k ( M ) diam( W ) k ) P R ( d r )= d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) E [ R d − k − j Ψ( α s R d − k )] . In the same way one shows that E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( − sλ d − k (Ξ) diam( W ) k )]= d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) E [ R d − k − j Ψ( − α s R d − k )] . Together with Theorem 3.2 this yields the result.
Remark 4.2.
It should be pointed out that there are only few examples of convex bodies for whichthe intrinsic volumes are available explicitly. For polytopes, they may be expressed in terms of thevolumes of lower-dimensional faces together with the external angle at these faces. For example, forthe cube one has that V j ([0 , d ) = (cid:18) dj (cid:19) , j ∈ { , , . . . , d } . On the other hand, for the d -dimensional unit ball B d one easily verifies that V j ( B d ) = κ d κ d − j (cid:18) dj (cid:19) , j ∈ { , , . . . , d } . In this section we assume that the window W is a general convex body in R d , but we strengthenthe assumptions on the typical cylinder base by assuming that Ξ arises from a fixed convex body M ⊂ R d − k by a uniform random rotation in R d − k , that is, we assume that Ξ =
U M , where U ∈ SO d − k is distributed according to the Haar measure ν d − k . Note that in this case m d − k = λ d − k ( M ) and p = 1 − e − γλ d − k ( M ) . Corollary 4.3.
Let the assumptions just described prevail. Then, one has P ( F − E F ≥ r ) ≤ exp (cid:18) rα − (cid:16) β + rα (cid:17) log (cid:18) rαβ (cid:19)(cid:19) , r ≥ , and P ( F − E F ≤ − r ) ≤ exp (cid:18) − rα − (cid:16) β − rα (cid:17) log (cid:18) − rαβ (cid:19)(cid:19) , ≤ r ≤ E F, where α = λ d − k ( M ) diam( W ) k and β = pλ d − k ( M ) d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) . (4.1)13 roof. We apply Corollary 4.1 and assume in addition that R = 1 P -almost surely. In this case pm d − k d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) E [ R d − k − j Ψ( α s R d − k )] = β Ψ( α s ) , and hence, P ( F − E F ≥ r ) ≤ exp (cid:18) inf s ≥ ( β Ψ( α s ) − rs ) (cid:19) = exp (cid:18) inf s ≥ ( β ( e αs − αs − − rs ) (cid:19) . It is easy to verify that the infimum is attained at s = α log(1 + rαβ ) . This gives P ( F − E F ≥ r ) ≤ exp (cid:16) rα − (cid:16) rα + β (cid:17) log (cid:16) rαβ (cid:17)(cid:17) for any r ≥ . This completes the proof for the upper tail.Similarly, consider the lower tail P ( F − E F ≤ − r ) ≤ exp (cid:18) inf s ≥ ( β Ψ( − α s ) − rs ) (cid:19) = exp (cid:18) inf s ≥ (cid:0) β ( e − αs + αs − − rs (cid:1)(cid:19) . In case r < αβ the infinum is attained at s = − α log(1 − rαβ ) and the proof is completed. It justremains to be justified that αβ ≥ E F for any convex M and W , and k ≥ .Due to the fact that intrinsic volumes are non-negative functionals on the family of convex bodies weconclude that αβ = p diam( W ) k d − k (cid:88) j =0 κ j κ d − j (cid:0) dj (cid:1) κ d V j ( W ) V d − k − j ( M ) ≥ p diam( W ) k κ d − k κ k (cid:0) dd − k (cid:1) κ d V d − k ( W )= p diam( W ) d κ d − k κ k (cid:0) dd − k (cid:1) κ d V d − k ( (cid:102) W ) , where (cid:102) W = diam( W ) − W and, thus, V d ( (cid:102) W ) ≤ . From the isoperimetric inequality for intrinsicvolumes of convex bodies (see, e.g., [16, Equation (14.31)]) we conclude that κ k (cid:0) dd − k (cid:1) V d − k ( (cid:102) W ) ≥ κ kd d V d ( (cid:102) W ) − kd ≥ κ kd d V d ( (cid:102) W ) . Substituting this into above inequality we get αβ ≥ p diam( W ) d κ d − k κ − kd d V d ( (cid:102) W ) = (cid:32) κ dd − k κ d − kd (cid:33) /d p V d ( W ) = (cid:32) κ dd − k κ d − kd (cid:33) /d E F. It remains to show that κ dd − k κ d − kd ≥ , which is equivalent to Γ (cid:18) d (cid:19) d ≥ Γ (cid:18) d − k (cid:19) d − k . (4.2)However, this follows from the fact that the function g ( x ) := Γ (cid:0) x (cid:1) /x , x > , is strictly increasingaccording to [14, Theorem 1]. This completes the proof.14 .3 Spherical windows Our general concentration inequality in Theorem 3.2 simplifies if we assume the shape of our observationwindow W to be spherical. More precisely, we assume that W = B dR is a centred Euclidean ball ofsome fixed radius R > . Corollary 4.4.
Let the general assumptions of Section 3 prevail, and let W = B dR . Assuming that E [ V j (Ξ) e sλ d − k (Ξ) ] < ∞ for some s > and all j ∈ { , , . . . , d − k } we have that, for r ≥ , P ( F − E F ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:104) pm d − k d − k (cid:88) j =0 R d − k − j κ d − k − j E [ V j (Ξ)Ψ( sλ d − k (Ξ)(2 R ) k )] − rs (cid:105)(cid:17) . Moreover, assuming that E [ V j (Ξ) λ d − k (Ξ)] < ∞ for all j ∈ { , , . . . , d − k } we have that, for ≤ r ≤ E F , P ( F − E F ≤ − r ) ≤ exp (cid:16) inf s ≥ (cid:104) pm d − k d − k (cid:88) j =0 R d − k − j κ d − k − j E [ V j (Ξ)Ψ( − sλ d − k (Ξ)(2 R ) k )] − rs (cid:105)(cid:17) . Proof.
We have to analyze the term E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )] appearing in Theorem 3.2, where now W = B dR . Since P d − k ( θ T B dR ) = B d − kR for any θ ∈ SO d,k andsince diam( B dR ) = 2 R we have that E [ λ d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( sλ d − k (Ξ) diam( W ) k )] = E [ λ d − k ( B d − kR + Ξ ∗ )Ψ( sλ d − k (Ξ)(2 R ) k )] . We are now in the position to apply Steiner’s formula [16, Equation (14.5)] in R d − k . Together withFubini’s theorem and the reflection invariance of the intrinsic volumes this yields E [ λ d − k ( B d − kR + Ξ ∗ )Ψ( sλ d − k (Ξ)(2 R ) k )] = d − k (cid:88) j =0 R d − k − j κ d − k − j E [ V j (Ξ)Ψ( sλ d − k (Ξ)(2 R ) k )] . This proves the claim for the upper tail, the lower tail is similar.If in addition the typical cylinder base is spherical as well, the inequalities simplify further. We assumethat P -almost surely Ξ = B d − kρ for some fixed ρ > . Then V j ( B d − kρ ) = κ d − k κ d − k − j (cid:18) d − kj (cid:19) ρ j , j ∈ { , , . . . , d − k } . Thus, d − k (cid:88) j =0 R d − k − j κ d − k − j E [ V j (Ξ)Ψ( sλ d − k (Ξ)(2 R ) k )]= κ d − k Ψ( sκ d − k ρ d − k (2 R ) k ) d − k (cid:88) j =0 (cid:18) d − kj (cid:19) R d − k − j ρ j = κ d − k Ψ( sκ d − k ρ d − k (2 R ) k ) R d − k (cid:16) ρR (cid:17) d − k . Putting a := κ d − k ρ d − k (2 R ) k and b := pm d − k κ d − k R d − k (1 + ρR ) d − k it is easy to check that the function f ( s ) := b (cid:0) e as − as − (cid:1) − rs attains its infimum over the set { s ≥ } at s = a log(1 + rab ) . Together with the previous corollary this yields the following result.15 orollary 4.5. If W = B dR and P -almost surely Ξ = B d(cid:37) for some fixed R, (cid:37) ∈ (0 , ∞ ) then P ( F − E F ≥ r ) ≤ exp (cid:16) ra − (cid:16) b + ra (cid:17) log (cid:16) rab (cid:17)(cid:17) , r ≥ , and P ( F − E F ≤ − r ) ≤ exp (cid:16) − ra − (cid:16) b − ra (cid:17) log (cid:16) − rab (cid:17)(cid:17) , ≤ r ≤ E F. Let us discuss the quality of the bounds we derived in the previous sections, where we restrict ourattention to Corollary 4.3. Since − rα − (cid:16) β − rα (cid:17) log (cid:18) − rαβ (cid:19) ≤ − r α β , ≤ r ≤ αβ, we infer for the lower tail that P ( F − E F ≤ − r ) ≤ exp (cid:16) − r α β (cid:17) , ≤ r ≤ E F. Next, we discuss the upper tail. For r → ∞ we obtain that P ( F − E F ≥ r ) ≤ exp( − Θ ( r log r )) , (4.3)where we recall that Θ ( r log r ) denotes a quantity in O ( r log r ) ∩ Ω ( r log r ) and our window W doesnot depend on r .Although no concentration inequality for F is explicitly available in the literature, such an inequalityeasily follows from the sharp cumulant estimates carried out in [5]. In fact, applying [15, Lemma 2.4]to these estimates yields a bound for the upper tail of the form P ( F − E F ≥ r ) ≤ exp( − Θ ( r )) , as r → ∞ . Clearly, this is weaker than the bound (4.3) we got. Moreover, if X is a Poisson randomvariable with parameter λ > then P ( X − E X ≥ r ) ≤ exp (cid:16) r − ( λ + r ) log (cid:16) rλ (cid:17)(cid:17) , r ≥ , which is asymptotically tight, as r → ∞ , up to a factor (2 π ( λ + r )) − / , see [9]. A comparison withCorollary 4.3 thus shows that, for a fixed window W , our bound for the upper tail is essentially ofthe same order as the one for a Poisson random variable. This leads us to the conclusion that theexponential order in r of our bound is presumably optimal.It is a remarkable observation that the bound (4.3) is of the same order as the one for the stationaryBoolean model in R d discussed in [2, 3]. This might be somewhat surprising, since the correlationstructure of the union set of a stationary Boolean model and of a stationary Poisson cylinder processare quite different. In fact, while for k = 0 the functional F is of volume-order, for k ≥ the randomset Z admits strong long-range correlations, which are propagated by the infinitely long cylinders overthe whole space. This is also well reflected, for example, by the growth of the variance of the totalvolume of Z for a sequence of growing windows W r = rW , r > . For example, it is known from [5]that the variance of λ d ( Z ∩ W r ) is of order r d + k , which for k ≥ is strictly larger than the volume-order r d . To relate this discussion to our inequalities, we shall now consider the case when the window isgrowing with r . In fact, we consider the situation in which the window is of the form r /d W for fixedconvex body W ⊂ R d . This choice corresponds to a linear growth of the volume of the window with16 . Moreover, we assume that the typical cylinder base arises from a fixed convex body M ⊂ R d − k bya uniform random rotation in R d − k . Then, recalling (4.1), we have that α = 2 k V d − k ( M ) r k/d and β = pm d − k d − k (cid:88) j =0 κ j r j/d V d − k − j ( M ) . We note that, as r → ∞ , α = Θ ( r k/d ) , while β = Θ ( r ( d − k ) /d ) . Plugging this into Corollary 4.3 wefind that P ( F − E F ≥ r ) ≤ exp( − Θ ( r − k/d )) , (4.4)as r → ∞ . This bound clearly reflects the dependence on the dimension parameter k and also showsthat the bound becomes weaker the bigger k is chosen. The purpose of this section is to prove a concentration inequality for the intrinsic volumes associatedwith the union set Z of a stationary and isotropic Poisson process of k -cylinders in R d . For this weassume in this section that the typical cylinder base Ξ is convex P -almost surely and also that thebase-direction distribution Q is rotation invariant. In view of Lemma 2.1 and the following discussion,this implies that Z is a stationary and isotropic random closed set. We also assume that the window W is convex. The proof of our tail bounds relies on the general concentration inequalities from Section 2.3 as well ason a mean value formula for the intrinsic volumes of Z ∩ W . While such formulas are well known forthe Boolean model (see, e.g., [16, Theorem 9.1.3]), we were not able to locate a corresponding resultfor the union set of Poisson cylinder processes in the existing literature (for the closest results in thisdirection we refer to [8, Section 5] and [20, Section 7]). The purpose of this section is to provide suchformulas under the assumption that Q is rotation invariant. In particular, this assumption allows usto use the principal kinematic formula for cylinders from [16, Chapter 6.3]. Proposition 5.1.
Let W ⊂ R d be convex body with V d ( W ) > and let ≤ j =: j ≤ d be someinteger. Suppose that Ξ is convex P -almost surely and that Q is rotation invariant. Assume furtherthat m i := E V i (Ξ) < ∞ for j − k ≤ i ≤ d − k . Then E V j ( Z ∩ W ) = ∞ (cid:88) (cid:96) =1 ( − (cid:96) − γ (cid:96) (cid:96) ! min( d,d + j − k ) (cid:88) j = j · · · min( d,d + j (cid:96) − − k ) (cid:88) j (cid:96) = j (cid:96) − c j (cid:96) j V j (cid:96) ( W ) (cid:96) (cid:89) i =1 c d + j i − − j i d m d − k + j i − − j i , where c pr = p ! κ p r ! κ r . If additionally j ≥ k , then E V j ( Z ∩ W ) = V j ( W ) (cid:0) − e − γ m d − k (cid:1) − e − γ m d − k d − j (cid:88) m =1 c m + jj V m + j ( W ) m (cid:88) p =1 ( − p γ p p ! (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i , where the empty sum is interpreted as zero. Remark 5.2.
We emphasize that for j = d and j = d − the formula in Proposition 5.1 canconsiderably be simplified. In fact, we have that E V d ( Z ∩ W ) = V d ( W )(1 − e − γm d − k ) , see (3.1), and E V d − ( Z ∩ W ) = γ V d ( W ) m d − k − e − γ m d − k + V d − ( W )(1 − e − γ m d − k ) . roof of Proposition 5.1. By definition of Z , the inclusion-exclusion principle and the multivariateMecke formula (see [11, Theorem 4.4]) we have that E V j ( Z ∩ W ) = E V j (cid:18) (cid:91) ( x,θ,K ) ∈ ξ Z ( x, θ, K ) ∩ W (cid:19) = ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) (cid:90) M (cid:96)d,k (cid:90) ( R d − k ) (cid:96) V j ( Z ( x , θ , K ) ∩ . . . ∩ Z ( x (cid:96) , θ (cid:96) , K (cid:96) ) ∩ W ) × λ (cid:96)d − k ( d ( x , . . . , x (cid:96) )) Q (cid:96) ( d (( θ , K ) , . . . , ( θ (cid:96) , K (cid:96) ))) . To evaluate the (cid:96) -fold integral over R d − k we make use of the following principal kinematic formula forcylinders, which can be found in [16, Corollary 6.3.1]. Namely, for fixed ( θ, K ) ∈ M d,k one has that (cid:90) SO d (cid:90) R d − k V j ( (cid:37)Z ( x, θ, K ) ∩ W ) λ d − k ( d x ) ν d ( d (cid:37) ) = min( d,d + j − k ) (cid:88) p = j c pj c d − p + jd V p ( W ) V d − k + j − p ( K ) , where ν d is the unique rotationally invariant Haar probability measure on SO d . A recursive applicationof this integral formula and Fubini’s theorem yields that, for fixed (cid:96) ∈ N and ( θ , K ) , . . . , ( θ (cid:96) , K (cid:96) ) ∈ M d,k , (cid:90) ( SO d ) (cid:96) (cid:90) ( R d − k ) (cid:96) V j ( (cid:37) Z ( x , θ , K ) ∩ . . . ∩ (cid:37) (cid:96) Z ( x (cid:96) , θ (cid:96) , K (cid:96) ) ∩ W ) × λ (cid:96)d − k ( d ( x , . . . , x (cid:96) )) ν (cid:96)d ( d ( (cid:37) , . . . , (cid:37) (cid:96) ))= min( d,d + j − k ) (cid:88) j = j · · · min( d,d + j (cid:96) − − k ) (cid:88) j (cid:96) = j (cid:96) − c j (cid:96) j V j (cid:96) ( W ) (cid:96) (cid:89) i =1 c d + j i − − j i d V d − k + j i − − j i ( K i ) , where we recall that j = j . Thus, from the assumed rotational invariance of Q and Fubini’s theoremwe conclude E V j ( Z ∩ W )= ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) (cid:90) M (cid:96)d,k (cid:90) ( SO d ) (cid:96) (cid:90) ( R d − k ) (cid:96) V j ( (cid:37) Z ( x , θ , K ) ∩ . . . ∩ (cid:37) (cid:96) Z ( x (cid:96) , θ (cid:96) , K (cid:96) ) ∩ W ) × λ (cid:96)d − k ( d ( x , . . . , x (cid:96) )) ν (cid:96)d ( d ( (cid:37) , . . . , (cid:37) (cid:96) )) Q (cid:96) ( d (( θ , K ) , . . . , ( θ (cid:96) , K (cid:96) )))= ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) min( d,d + j − k ) (cid:88) j = j · · · min( d,d + j (cid:96) − − k ) (cid:88) j (cid:96) = j (cid:96) − c j (cid:96) j V j (cid:96) ( W ) (cid:96) (cid:89) i =1 c d + j i − − j i d m d − k + j i − − j i . This proves the first claim.If j ≥ k , the above formula can be simplified further. To this end, let us introduce the notation q i := j i − j i − . Then (cid:96) (cid:80) i =1 q i = j (cid:96) − j and we obtain that E V j ( Z ∩ W )= ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) d (cid:88) j = j · · · d (cid:88) j (cid:96) = j (cid:96) − c j (cid:96) j V j (cid:96) ( W ) (cid:96) (cid:89) i =1 c d + j i − − j i d m d − k + j i − − j i ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) d − j (cid:88) q =0 d − j − q (cid:88) q =0 · · · d − j − q − ... − q (cid:96) − (cid:88) q (cid:96) =0 c q + ... + q (cid:96) + j j V q + ... + q (cid:96) + j ( W ) (cid:96) (cid:89) i =1 c d − q i d m d − k − q i = V j ( W )(1 − e − γm d − k ) + d − j (cid:88) m =1 c m + jj V m + j ( W ) S m with S m := ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) (cid:88) q ,...,q (cid:96) ≥ q + ... + q (cid:96) = m (cid:96) (cid:89) i =1 c d − q i d m d − k − q i . Assume further that (cid:96) = r + p , ≤ p ≤ m , r ∈ { , , , . . . } and q , . . . , q p ∈ N , q p +1 = . . . = q (cid:96) = 0 .Then the infinite sum S can be evaluated explicitly. Indeed, we have that S m = ∞ (cid:88) (cid:96) =1 ( − (cid:96) − (cid:96) ! γ (cid:96) (cid:88) q ,...,q (cid:96) ≥ q + ... + q (cid:96) = m (cid:96) (cid:89) i =1 c d − q i d m d − k − q i = m (cid:88) p =1 ∞ (cid:88) r =0 ( − r + p − γ r + p ( r + p )! (cid:18) r + pr (cid:19) m rd − k (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i = − e − γm d − k m (cid:88) p =1 ( − p γ p p ! (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i and the proof is complete. For fixed j ∈ { , , . . . , d } we consider the Poisson functional F j := V j ( Z ∩ W ) . We start by dealing with the first-order difference operator of F j . Due to the additivity of the intrinsicvolumes, for ( λ d − k ⊗ Q ) -almost all ( x, θ, K ) ∈ R d − k × M d,k we have that D ( x,θ,K ) F j = V j (( Z ∪ Z ( x, θ, K )) ∩ W ) − V j ( Z ∩ W )= V j ( Z ( x, θ, K ) ∩ W ) − V j ( Z ∩ Z ( x, θ, K ) ∩ W ) holds P -almost surely.In order to derive a bound for the upper tail and the lower tail of the functional F j we will apply thetechnique already used in Section 3 for the case of the volume. For this we need to make sure that theconditions of Lemma 2.2 hold for F j . Lemma 5.3.
For any j ∈ { , , . . . , d } we have that F j ∈ L ( P ) , DF j ∈ L ( P ⊗ λ d − k ⊗ Q ) and s F j = s (lt) F j = ∞ .Proof. Since the intrinsic volumes V j are non-negative and monotone under set inclusion on the familyof convex bodies we have that P -almost surely D ( x,θ,K ) F j ≤ V j ( Z ( x, θ, K ) ∩ W ) ≤ V j ( W ) for all ( x, θ, K ) ∈ R d − k × M d,k . The rest of the proof is now analogous to the proof of Lemma 3.1.19 heorem 5.4. Let W ⊂ R d be a convex body with V d ( W ) > , Ξ be convex P -almost surely and assumethat Q is rotation invariant. Also, suppose that j ≥ k and m i ∈ (0 , ∞ ) for all j − k ≤ i ≤ d − k . Then,for all r ≥ , one has that P ( F j − E F j ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) E (cid:104) V d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ (cid:16) s min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:0) kj − i (cid:1) V i (Ξ) (cid:17) × d − j (cid:88) m =0 β m (cid:16) min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:0) kj − i (cid:1) V i (Ξ) (cid:17) m/j (cid:105) − rs (cid:17)(cid:17) , and for ≤ r ≤ E F j one has that P ( F j − E F j ≤ − r ) ≤ exp (cid:16) inf s ≥ (cid:16) E (cid:104) V d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ (cid:16) − s min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:0) kj − i (cid:1) V i (Ξ) (cid:17) × d − j (cid:88) m =0 β m (cid:16) min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:0) kj − i (cid:1) V i (Ξ) (cid:17) m/j (cid:105) − rs (cid:17)(cid:17) , where Ψ( x ) = e x − x − , x ∈ R , and β := 1 − e − γm d − k m d − k , β = 0 ,β m := κ m/jd − j (cid:0) dj + m (cid:1) c m + jj κ m/jd κ d − j − m (cid:0) dj (cid:1) m/j (cid:98) m (cid:99) (cid:88) p =1 m − p − d − k (cid:16) − e − γm d − k p (cid:88) i =0 ( γm d − k ) p − i (2 p − i )! (cid:17) × (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i for m ∈ { , . . . , d − j } . Remark 5.5. (i) We specialize the result of Theorem 5.4 for j = d and j = d − , where theconcentration inequality takes a more simple form. For simplicity, we restrict ourselves to thebound for the upper tail. If j = d we obtain, for r ≥ , P ( F d − E F d ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d − k E (cid:104) V d − k ( P d − k (Θ T W ) + Ξ ∗ )Ψ( s diam( W ) k V d − k (Ξ)) (cid:105) − rs (cid:17)(cid:17) , which is precisely the bound we derived in Section 3 under more general conditions, since V d − k ( K ) = λ d − k ( K ) for a convex body K ⊂ R d − k . Moreover, choosing j = d − we obtain,again for r ≥ , P ( F d − − E F d − ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d − k E (cid:104) V d − k ( P d − k (Θ T W ) + Ξ ∗ ) × Ψ (cid:0) s diam( W ) k − [diam( W ) V d − k − (Ξ) + kV d − k (Ξ)] (cid:1)(cid:105) − rs (cid:17)(cid:17) . (ii) Taking k = 0 , which corresponds to the Boolean model, and j = d − we deduce that P ( F d − − E F d − ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) pm d E (cid:2) V d ( W + Ξ ∗ )Ψ( sV d − (Ξ)) (cid:3) − rs (cid:17)(cid:17) , r ≥ , which should be compared to the corresponding inequality (3.2) for F d . Note that the reasonbehind this simple form is the fact that the constant β in Theorem 5.4 is equal to zero. Since this20s not the case for β m with m ∈ { , . . . , d − j } , the resulting inequalities become more involved.In fact, for j ∈ { , , . . . , d − } we have that P ( F j − E F j ≥ r ) ≤ exp (cid:16) inf s ≥ (cid:16) E (cid:104) V d ( W + Ξ ∗ )Ψ( sV j (Ξ)) d − j (cid:88) m =0 β m V j (Ξ) m/j (cid:105) − rs (cid:17)(cid:17) , r ≥ . Proof of Theorem 5.4.
As in the proof of Theorem 3.2 we start by deriving an upper bound for thefunction V F j ( s ) defined by (2.6). Considering the term T t , t ∈ [0 , , and applying Proposition 5.1 wehave that, putting Λ := γ λ d − k ⊗ Q , T t ≤ V j ( Z ( x, θ, K ) ∩ W ) − (cid:90) N V j ( Z ( µ ) ∩ Z ( x, θ, K ) ∩ W ) Π (1 − t )Λ ( d µ )= e − γ (1 − t ) m d − k (cid:18) V j ( Z ( x, θ, K ) ∩ W )+ d − j (cid:88) m =1 c m + jj V m + j ( Z ( x, θ, K ) ∩ W ) m (cid:88) s =1 ( − γ (1 − t )) s s ! (cid:88) q ,...,q s > q + ... + q s = m s (cid:89) i =1 c d − q i d m d − k − q i (cid:19) , where for µ ∈ N , Z ( µ ) stands for the union set induced by µ . Hence, (cid:90) T t d t = V j ( Z ( x, θ, K ) ∩ W ) I + d − j (cid:88) m =1 c m + jj V m + j ( Z ( x, θ, K ) ∩ W ) × m (cid:88) p =1 I p γ p (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i , where I p : = 1 p ! (cid:90) ( t − p e γ ( t − m d − k d t = ( − p ( γm d − k ) p +1 (cid:32) − e − γm d − k p (cid:88) i =0 ( γm d − k ) p − i ( p − i )! (cid:33) . Let us introduce the following additional notation in order to simplify our subsequent computations: α := 1 − e − γm d − k m d − k , α = 0 ,α m := c m + jj (cid:98) m (cid:99) (cid:88) p =1 m − p − d − k (cid:32) − e − γm d − k p (cid:88) i =0 ( γm d − k ) p − i (2 p − i )! (cid:33) (cid:88) q ,...,q p > q + ... + q p = m p (cid:89) i =1 c d − q i d m d − k − q i for m ≥ . Using the notation above and applying Fubini’s theorem and the fact that intrinsic volumesare non-negative functionals on the family of convex bodies we conclude that V F ( s ) ≤ v ( s ) , s ≥ , with v ( s ) given by v ( s ) = (cid:90) M d,k (cid:90) R d − k (cid:16) e sV j ( Z ( x,θ,K ) ∩ W ) − (cid:17) d − j (cid:88) m =0 α m V m + j ( Z ( x, θ, K ) ∩ W ) λ d − k ( d x ) Q ( d ( θ, K )) .
21n the next step we investigate the integral w ( s ) := s (cid:90) v ( u ) d u. For that purpose we notice that in the definition of w ( s ) we can multiply the integrand with theindicator function { Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ } . In fact, Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ is equivalent to V j ( Z ( x, θ, K ) ∩ int( W )) > and, additionally, λ d − k ( { x ∈ R d − k : Z ( x, θ, K ) ∩ W (cid:54) = ∅ and Z ( x, θ, K ) ∩ int( W ) = ∅ } ) = 0 holds by our convexity assumption on the cylinder bases K . We can thus write w ( s ) : = s (cid:90) v ( u ) d u = (cid:90) M d,k (cid:90) R d − k (cid:104) e sV j ( Z ( x,θ,K ) ∩ W ) − sV j ( Z ( x, θ, K ) ∩ W ) − (cid:105) × d − j (cid:88) m =0 α m V m + j ( Z ( x, θ, K ) ∩ W ) V j ( Z ( x, θ, K ) ∩ W ) { Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ } λ d − k ( d x ) Q ( d ( θ, K ))= (cid:90) M d,k (cid:90) R d − k Ψ( sV j ( Z ( x, θ, K ) ∩ W )) d − j (cid:88) m =0 α m V m + j ( Z ( x, θ, K ) ∩ W ) V j ( Z ( x, θ, K ) ∩ W ) × { Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ } λ d − k ( d x ) Q ( d ( θ, K )) , where Ψ( x ) = e x − x − , x ∈ R . Let us note here that the function Ψ( x ) is increasing for x ≥ and that all coefficients α m are non-negative. Thus, the integrand is an increasing function in V m + j ( Z ( x, θ, K ) ∩ W ) , ≤ m ≤ d − j . From the isoperimetric inequalities for intrinsic volumes ofconvex bodies (see, e.g., [16, Equation (14.31)]) we deduce that, for ( x, θ, K ) ∈ M d,k , V m + j ( Z ( x, θ, K ) ∩ W ) ≤ κ m/jd − j (cid:0) dj + m (cid:1) κ m/jd κ d − j − m (cid:0) dj (cid:1) m/j V j ( Z ( x, θ, K ) ∩ W ) m/j +1 . Provided that Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ , this implies w ( s ) ≤ (cid:90) M d,k (cid:90) R d − k Ψ( sV j ( Z ( x, θ, K ) ∩ W )) d − j (cid:88) m =0 β m V j ( Z ( x, θ, K ) ∩ W ) m/j λ d − k ( d x ) Q ( d ( θ, K )) , (5.1)with the coefficients β , . . . , β d − j given by β m := α m κ m/jd − j (cid:0) dj + m (cid:1) κ m/jd κ d − j − m (cid:0) dj (cid:1) m/j , m ∈ { , , . . . , d − j } . Note that we can from now on omit the indicator function that Z ( x, θ, K ) ∩ int( W ) (cid:54) = ∅ . Using againthe fact that the window W as well as the cylinder bases K are convex and that the intrinsic volumesare monotone under set inclusion on the family of convex bodies we get V j ( Z ( x, θ, K ) ∩ W ) ≤ V j ( K + diam( W ) C k ) , where C k ⊂ E k denotes the k -dimensional unit cube. Applying now [16, Lemma 14.2.1] we concludethat V j ( Z ( x, θ, K ) ∩ W ) ≤ min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:18) kj − i (cid:19) V i ( K ) . w ( s ) ≤ (cid:90) M d,k V d − k ( P d − k ( θ T W ) + K ∗ ) Ψ s min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:18) kj − i (cid:19) V i ( K ) × d − j (cid:88) m =0 β m min { d − k,j } (cid:88) i = j − k diam( W ) j − i (cid:18) kj − i (cid:19) V i ( K ) m/j Q ( d ( θ, K )) . This completes the proof for the upper tail, the proof for the lower tail is similar.
The result of Theorem 5.4 can be simplified further if we additionally assume that the cylinder basesare random rotations of a fixed convex body M ⊂ R d − k and the direction Θ of the typical cylinderbase is uniformly distributed on SO d,k according to the unique rotationally invariant Haar probabilitymeasure ν d,k on SO d,k , independently of U (recall Sections 4.1 and 4.2). More explicitly, this meansthat Ξ =
U M , where U ∈ SO d − k is a uniform random rotation in R d − k , and that Ξ and Θ areindependent. Corollary 5.6.
Under the assumptions just described we have that, for all integers k ≤ j ≤ d , P ( F j − E F j ≥ r ) ≤ exp (cid:18) rα − (cid:16) β + rα (cid:17) log (cid:18) rαβ (cid:19)(cid:19) , r ≥ , and P ( F j − E F j ≤ − r ) ≤ exp (cid:18) − rα − (cid:16) β − rα (cid:17) log (cid:18) − rαβ (cid:19)(cid:19) , ≤ r ≤ E F j , where α = min { d − k,j } (cid:80) i = j − k diam( W ) j − i (cid:0) kj − i (cid:1) V i ( M ) and β = d − k (cid:80) i =0 κ i κ d − i ( di ) κ d V i ( W ) V d − k − i ( M ) d − j (cid:80) m =0 β m α m/j , with β m defined as in Theorem 5.4.Proof. The proof is analogous to the proof of Corollary 4.1 and Corollary 4.3 due to invariance ofintrinsic volumes under rotations.
Remark 5.7.
As in Section 4.4 we consider the special case when the window is of the form r /d W forsome fixed convex body W ⊂ R d . For this, we fix j ∈ { k, . . . , d − } and use that log(1 + x ) behaveslike x for small values of x . Then one can easily check that, as r → ∞ , P ( F j − E F j ≥ r ) ≤ exp (cid:0) − Θ ( r − k/d ) (cid:1) , which is independent of j . This should be compared to the bound (4.4) for the volume in this situation. Acknowledgement
We would like to thank the other members of our team for stimulating discussions about the topicof this paper during our regular research seminars in the summer term 2019. We also thank ClaudiaRedenbach (Kaiserslautern) for providing the two simulations shown in Figure 1.1 and Günter Last(Karlsruhe) for encouraging us to study the case of expanding windows.A.B. and A.G. were supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131
High-dimensional Phenomena in Probability – Fluctuations and Discontinuity . C.T. was supported by theDFG Scientific Network
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