Exponential Moments for Planar Tessellations

In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical tessellations such as the Poisson–Voronoi, Poisson–Delaunay and Poisson line tessellation, we also treat the Johnson–Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

point process is given by a Poisson point process (PPP), it is usually possible to derive first and second moments for these characteristics as a function of the intensity λ, see [16, [12][13][14]. However, to derive complete and tractable descriptions of the whole distribution of these characteristics is often difficult.
In this paper we contribute to this line of research by proving existence of all exponential moments for the distribution of the total edge length in a unit disk. More precisely, let B r ⊂ R 2 denote the closed centered disk with radius r > 0 and let |S ∩ A| = ν 1 (S ∩ A) denote the random total edge length of the tessellation S ⊂ R 2 in the Lebesgue measurable volume A ⊂ R 2 , where ν 1 denotes the one-dimensional Hausdorff measure. We show for a large class of tessellations that for all α ∈ R we have that As a motivation, let us mention that the information on the tail behavior of the distribution of |S ∩ B 1 | provided by (1)  exists, see [10,Lemma 6.1]. This can be used for example to establish the limiting behavior of the percolation probability for the Boolean model with large radii based on Cox point processes where the intensity measure is given by |S ∩ dx|, see [10]. Moreover, existence of exponential moments plays a role in establishing percolation in an SINR graph based on Cox point processes in the case of an unbounded integrable path-loss function, see [17] for details.

Tessellations
Let ∂ A =Ā \ A o denote the boundary of a set A ⊂ R 2 and write x = (x 1 , x 2 ) for x ∈ R 2 . Apart from the classical Voronoi tessellation (VT), where and its dual, the Delaunay tessellation (DT), where 1] we also consider the line tessellation (LT), where {x ∈ R 2 : x 1 cos X i,2 + x 2 sin X i,2 = X i,1 }.
See Fig. 1 for realizations of the VT and the DT and their intersections with B 1 in case X is a homogeneous PPP. The extension of the VT known as the Johnson-Mehl tessellation (JMT) is covered by our results, see for example [1]. For this consider the i.i.d. marked stationary point process 0 where we use the same notation | · | for the Euclidean norm on R 2 and [0, ∞). Then, the JMT is given by 0), (X j , T j ))}.
We also consider the Manhattan grid (MG), see for example [9]. For this let Y = (Y v , Y h ) be the tuple where Y v = {Y i,v } i∈I v and Y h = {Y i,h } i∈I h are two independent simple stationary point processes on R. Then the MG is defined as Note that S M is stationary, similarly to all previously defined tessellations, however, unlike them, it is not isotropic. One can make S M isotropic by choosing a uniform random angle in [0, 2π), independent of Y , and rotating S M by this angle. Our results for the MG will be easily seen to hold for both the isotropic and anisotropic version of the MG.
Next, let us denote by (C i ) i∈J the collection of cells in the tessellation S, where J = J (S). Formally, a cell C i of S is defined as an open subset of R 2 such that C i ∩ S = ∅ and ∂C i ⊂ S. In view of applications, see for example [9] or [15], it is sometimes desirable to consider nested tessellations (NT), which we can partially treat with our techniques. For this, let S o be one of the tessellation processes introduced above, defined via the point process X (o) , with cells (C i ) i∈J , which now serves as a first-layer process. For every i ∈ J , let S i be an independent copy of one of the above tessellation processes, maybe of the same type as S o with potentially different intensity or maybe of a different type, but all S i should be of the same type and have the same intensity. Let X (i) denote the underlying independent point process of S i . Then the associated NT is defined as Here, i∈J (S i ∩ C i ) will be called the second-layer tessellation. This definition of a NT originates from [19,Section 3.4.4], where this class of tessellations was defined as a special case of iterated tessellations.
Note that all kinds of tessellations S defined in this section are stationary, i.e., S equals S + x in distribution for any given x ∈ R 2 . However, for a planar tessellation in order to be stationary, it is not required that it is based on a stationary point process. For example, let Y be a homogeneous Poisson process in R, and let X = {(X i , 0) : X i ∈ Y }. Then X is not a stationary point process in R 2 , however, the associated process {(X i , t) : X i ∈ Y , t ∈ R} of infinite vertical lines is a stationary tessellation.
Finally note that all subgraphs of tessellations having the property (1) inherit this property by monotonicity. In particular, our results cover the cases of the Gabriel graph, the relative neighborhood graph, and the Euclidean minimum spanning tree, since they are subgraphs of the DT, presented in decreasing order with respect to inclusion.

Assumptions
Unless noted otherwise, throughout the manuscript X = {X i } i∈I denotes a stationary point process on R 2 with intensity 0 < λ < ∞. Our results will use the following assumptions on exponential moments for the number of points and void probabilities for the underlying stationary point process. First, for the VT, we assume that for all β > 0. Second, we assume that We provide the easy proof that these conditions hold for the homogeneous PPP in Sect. 1.4. They can also be verified for example for some b-dependent Cox point processes and some Gibbsian point processes, see also Sect. 1.4. For the JMT, we generally assume that the mark distribution μ(dt) is absolutely continuous with respect to the Lebesgue measure. Further, let B J r denote the centered ball in the JM metric as defined in (2). Then, in analogy to the above, we assume that for all β > 0. Second, we assume that Again, these conditions hold if (X i ) i∈I is homogeneous PPP and μ is for example the Lebesgue measure, see Sect. 1.4. For the LT, we will assume that there exists β ≤ ∞ such that the random variable #(X ∩ ([−1, 1] × [0, 2π])) has exponential moments up to β , i.e., for all β < β . This condition holds for example for the homogeneous PPP with β = ∞.
For the MG, we assume that there exist β v , β h ≤ ∞ such that the random variables #(Y v ∩ [0, 1]) and #(Y h ∩ [0, 1]) have all exponential moments up to β v , β h , i.e., for all β < β v , respectively β < β h . This condition is satisfied with β v = ∞ if Y v is a homogeneous Poisson process, and analogously for β h .

Results
Having defined the types of tessellations we consider, we can now state our main theorem with its proof and all other proofs presented in Sect. 2. Note that, using Hölder's inequality and stationarity, the statement of Theorem 1.1 and all subsequent results remain true if B 1 is replaced by any bounded measurable subset of R 2 .
Let us briefly comment on the proof of Theorem 1.1. The proof of the parts for the LT and MG is rather straightforward. As will become clear from the proof, in case of the MG, an application of Hölder's inequality would give the same result without the independence assumption on the point processes Y v , Y h , but we lose some of the exponential moments. The cases for the VT, JMT and DT are more involved. However, the statements follow easily if exponential moments for the corresponding number of edges intersecting B 1 can be established. More precisely, let (E i ) i∈K denote the collection of edges in the tessellation S, where K = K (S), and holds for all α > 0, then so does (1) for all α > 0. If (10) holds for some α > 0, then so does (1) for some α > 0. The following result establishes exponential moments for W and also the simple consequence that for some α > 0, where is the number of cells intersecting B 1 .  (4), respectively (5) and (6), for all β > 0, (11) holds for all α ∈ R. For Delaunay tessellations based on a homogeneous Poisson point process, (11) holds for some α > 0.
As mentioned above, Theorem 1.1 parts (i) and (ii) are immediate consequences of Proposition 1.2 part (i) for the corresponding tessellations. However, for the case of the DT, as in part (iii) of Theorem 1.1, we cannot use Proposition 1.2 since we do not have a statement for all α > 0. In order to overcome this difficulty, we first estimate small exponential moments of the total number of edges intersecting with B a for different values of a > 0 and then use an additional scaling argument to conclude (1) for all α > 0. Let us also emphasize that for the DT, we establish the above results only in the case in which the underlying point process is a PPP. It is unclear if exponential moments for the number of edges W and number of cells V intersecting with the unit disk exist for the LT and we make no statements about them. For the NT, existence of exponential moments for V for the first-layer tessellation can be used to verify (1) for S N . More precisely, we have the following result.

Corollary 1.3 Consider the nested tessellation.
(i) If for the first-layer tessellation (11) holds for all α ∈ R and for the second-layer tessellation (1) holds for all α ∈ R, then also S N satisfies (1) for all α ∈ R. (ii) If for the first-layer tessellation (11) holds for some α > 0 and for the second-layer tessellation (1) holds for some α > 0, then also S N satisfies (1) for some α > 0.
As we will explain in Sect. 1.5, the statement of Proposition 1.2 is false for the MG based on independent homogeneous Poisson processes on the axes, despite the fact that (1) holds in this case according to Theorem 1.1. However, in the special case where the NT is composed of MGs in both layers and the second-layer MG is based on independent homogeneous Poisson processes, for this S N , we still obtain (1) for all α ∈ R. This is the content of the following result. (1) for all α ∈ R. Then, (1) holds for the nested Manhattan grid also for all α ∈ R. 0

Proposition 1.4 Consider the nested tessellation and assume that the second-layer tessellation is given by Manhattan grids based on two independent homogeneous Poisson processes and the first-layer tessellation is also a Manhattan grid satisfying
Let us mention that for the tessellations studied in Theorem 1.1, considering Palm versions of the underlying point process, at least in the case where it is a homogeneous PPP, does not change existence of all exponential moments. We want to be precise here since there are multiple different possibilities to define Palm measures in this context. For the Poisson-VT, Poisson-JMT and Poisson-DT, we denote by X * the Palm version of the underlying unmarked PPP and denote by S * = S(X * ) its associated tessellations. For the Poisson-LT we denote by X * the Palm version of the underlying PPP only with respect to the first coordinate, i.e., X * = X ∪ {(0, Φ)}, where Φ is a uniform random angle in [0, π) that is independent of X . Roughly speaking, this corresponds to S * L = S L (X * ) being distributed as S L when conditioned to have a line crossing the origin o of R 2 with no fixed angle. The Palm version of the MG is given by where U is an independent uniformly distributed random variable on [0, 1] and Y * v and Y * h denote the Palm versions of Y v and Y h , see [9, Section III.B]. We will recall the notion of the Palm version of a general stationary point process in Sect. 2.3. Palm distributions of NTs can be defined correspondingly, see for example [9,19].

Corollary 1.5 Consider all the tessellations S appearing in Theorem 1.1. If the underlying point processes are homogeneous Poisson point processes, we also have for all
To end this section with a short discussion, let us mention that it is a simple consequence of the works [2,8,20] that for all α ∈ R where N * denotes the number of Poisson-Delaunay edges originating from the origin under the Palm distribution for the underlying PPP. The assertion (15) seems similar to the one (14) for the Poisson-DT, however, S * ∩ B 1 can contain segments from many edges that are not adjacent to the origin, in particular also from edges both endpoints of which are situated outside B 1 . It is an interesting open question whether it is possible to provide a simpler proof of the assertion (14) for all α ∈ R or the assertion (1) for all α ∈ R for the Poisson-DT based on the fact that (15) holds for all α ∈ R. For Corollary 1.5, we provide a case-by-case proof. Let us mention that, for the reverse implication with S * = S(X * ) for a homogeneous PPP X with intensity λ, using the inversion formula of Palm calculus [11,Section 9.4] and Hölder's inequality, we can derive the following criterion, where C o is the Voronoi cell of the origin in the VT S V (X * ).
Finally, let us comment on possible generalizations of our results to higher dimensions. In at least three dimensions, it is still true that VTs, DTs and JMTs are exponentially stabilizing, i.e., the probability that a point of the underlying point process outside the ball B k influences the realization of the tessellation inside B 1 decays exponentially fast, which is an important argument in our proofs. However, in the planar case, given that the points inside B k determine the tessellation inside B 1 , the total number of edges intersecting with B 1 can be bounded by constant times the number of points in the region B k (see e.g. Sect. 2.1.1 for details). This is in general not true in higher dimensions, which yields the main obstacle for generalizing our results. On the other hand, some of our results extend easily to higher dimensions. For example, defining a higher-dimensional analogue of a Manhattan grid using independent stationary point processes on all coordinate axes and connecting all these points by edges, an analogue of Theorem 1.1 can easily be derived using arguments similar to the ones of Sect. 2.1.5.

Examples: Poisson-, Cox-and Gibbs-Voronoi Tessellations
It is easy to check that the assumptions listed in Sect. 1.2, are satisfied if the underlying point process is a stationary PPP. Indeed, for (3) note that by the Laplace transform, for any Further, for (4) note that the void probability for the PPP is given by As for the assumptions (5) and (6), the same arguments can be applied.
It is natural to ask under what conditions existence of exponential moments for the total edge length in the unit disk can be guaranteed for tessellations S(X ) where X is not a PPP but some different stationary planar point process. As a starting point for future studies, in this section we present examples for the VT based on a stationary Cox point process (CPP) and a stationary Gibbsian point process (GPP) X where our results guarantee the existence of exponential moments.

Cox-Voronoi Tessellations
A Cox point process is a PPP with random intensity measure Λ(dx), see for example [6] for details. We have the following proposition.
for all β > 0. Then, for S = S V (X ), (1) holds for all α ∈ R. and thus (16) is precisely what we need. The same argument can be applied for assumption (3). 0 The conditions (16) and (17) hold if Λ(Q 1 ) has all exponential moments and Λ is bdependent, where we call Λ b-dependent if for any two measurable sets A, B ⊂ R 2 such that dist(A, B) = inf x∈A,y∈B |x − y| > b, the restrictions Λ| A and Λ| B of Λ to A respectively B are independent. Indeed, by stationarity of Λ, it suffices to verify (16) and (17) with B k replaced by Q k = [−k/2, k/2] 2 (both for k = n and k = n + 4) in the limit N k → ∞. Let us assume that Λ is b-dependent. Then, there exists b = b (b) ∈ N such that for any k ∈ N, Q k can be partitioned into at most b disjoint subsets such that each of these subsets consists of (apart from the boundaries) disjoint copies of Q 1 such that the restrictions of Λ to these copies are mutually independent. Using this independence and the existence of all exponential moments of Λ(Q 1 ), further applying Hölder's inequality for the collection of partition sets, (16) and (17) follow. A relevant example for a b-dependent and even bounded intensity measure is the modulated Poisson point process where Λ(dx) = dx(λ 1 1{x ∈ Ξ } + λ 2 1{x ∈ Ξ c }), with Ξ being a Poisson-Boolean-model with bounded radii, and λ 1 , λ 2 ≥ 0, see [5,Section 5.2.2]. Another example for which conditions (16) and (17) holds, and which is unbounded, is the shot-noise field, see [5,Section 5.6], where Λ(dx) = dx i∈I κ(x − Y i ) for some integrable kernel κ : R 2 → [0, ∞) with compact support and {Y i } i∈I a stationary PPP.

Gibbs-Voronoi Tessellations
A Gibbs point process on R 2 is defined via its conditional probabilities in bounded measurable volumes B ⊂ R 2 . They take the form of a Boltzmann weight where P B is a PPP on B with intensity λ > 0, γ ∈ R is a system parameter and H is the Hamiltonian, which assigns some real-valued energy to the configuration X B X B c = X B ∪ X B c , where X B c is a boundary configuration in B c = R 2 \ B. For details see for instance [7].
As an example, we consider the Widom-Rowlinson model where H (X ) = | X i ∈X B r (X i )|, with B r (x) the ball of radius r > 0, centered at x ∈ R 2 . Existence of associated point processes on R 2 that are stationary can be guaranteed, see for example [4]. We have the following result.  = λ(e β − e −γ πr 2 ) < ∞, for all r , λ, γ, β > 0, which proves the desired result.

Absence of Exponential Moments for the Number of Edges and Cells
In Proposition 1.2, we provide statements about existence of exponential moments for V , the number of cells intersecting B 1 , and W , the number of edges intersecting B 1 . In this section we want to exhibit one example in our family of tessellations for which exponential moments for V do not exist. Indeed, take the MG where the underlying stationary point processes are PPPs Y v and Y h with intensity λ. By translation invariance, we can also consider the random variable V , the number of cells intersecting Q 1 . In order to simplify the notation, let us write . These random variables are independent and Poisson distributed with parameter λ. Then we have that Since for the MG based on PPPs, V and the number W ∞ of infinite lines intersecting with Q 1 are of the same order, further, W ∞ ≤ W , it follows that E[exp(αW )] = ∞.

Proofs
For our results, it obviously suffices to consider α > 0 instead of α ∈ R.

Total Edge Length, Number of Edges and Cells: Proof of Theorem 1.1 and Proposition 1.2
The proof of Theorem 1.
where we recall that (E i ) i∈K is the collection of edges of S V . The following lemma states that unless we have a void space, numbers of edges can be bounded from above by numbers of points in bounded regions.
Proof Let us assume existence of X i ∈ X ∩ B b . We first claim that for any edge of S V intersecting with B a , the corresponding edge in the dual DT connects two points in B b+3a . Indeed, assume otherwise, then there exists v ∈ B a and X j ∈ X ∩ B c b+3a such that |v − X j | = min{|v − X l |: l ∈ I } and On the other hand, which is a contradiction. Thus, for any Voronoi edge intersecting with B a ⊆ B b , the corresponding Delaunay edge has both endpoints in X ∩ B b+3a . But since the subgraph of the Delaunay graph spanned by the vertex set X ∩ B b+3a is simple and planar, Euler's formula (see e.g. [12, Remark 2.1.4]) implies that the number of such edges is bounded by 3 times the number of vertices in this subgraph. This implies (19).
Note that Lemma 2.1 holds for any point cloud X . The proof of (10) for the VT now rests on the assumption that it is exponentially unlikely to have large void spaces of order k 2 and existence of exponential moments for numbers of points in annuli of order k. Let denote the distance of the closest point in X to the origin.

Proof of Proposition 1.2 part (i)
In the event {R ≤ 1} we have that B 1 ∩ X = ∅, and therefore by Lemma 2.1 applied for a = b = 1, we obtain On the other hand, in the event {R ≥ 1}, we can apply Lemma 2.1 with a = 1 and b = R in order to obtain that, almost surely, where we also used that by stationarity, on ∂ B R there is precisely one point, almost surely. By assumption (4) we have E[R] < ∞ and hence P(R < ∞) = 1. We can thus estimate for all α > 0, where we used Hölder's inequality in the last line. Now, by the assumptions (4) and (3), there exist c 1 , c 2 > 0 such that for sufficiently large k, we have and hence summability of the right-hand side of (21) is guaranteed. This concludes the proof of Proposition 1.2 for the number of edges and thus of Theorem 1.1 part (i).

Johnson-Mehl Tessellations: Proof of Proposition 1.2 Part (i)
As explained in Sect. 1, Theorem 1.1 (ii) follows once we verify (10) Proof Assume that there exists i ∈ I such that (X i , T i ) ∈ B J b and that S J exhibits an edge having a non-empty intersection with B a , and let x ∈ B a be a point of such an edge. Then, using the triangle inequality, since and for any j ∈ I with (X j , and the result follows. Proof of (10) for the JMT for all α > 0. We start with two preliminary observations. First, let E denote the set of (closed) edges of S J . By construction of a JMT, almost surely, any 0 E ∈ E has the property that there exist precisely two points (X i , T i ), (X j , T j ) (depending on E) such that for all z ∈ E d J ((z, 0), , 0), (X k , T k )).
In this case, we will write E = (X i , T i ); (X j , T j ) . We claim that for any finite subset I 0 of I , holds. Indeed, the set on the left-hand side of (22) is in one-to-one correspondency with #D(I 0 ) where if and only if X i and X j are connected by an edge in the dual of the Johnson-Mehl graph. Note that since JMT is a planar graph, so is its dual, and thus D(I 0 ) has cardinality at most 3#I 0 thanks to the Euler formula for planar graphs. Now, let us define the distance of the closest point to the (space-time) origin in the Johnson-Mehl metric Now, in the event {R ≤ 1}, we have B J 1 ∩ X = ∅, and thus an application of Lemma 2.2 for Thanks to (22), the right-hand side is at most #( X ∩ B J 4 ). On the other hand, in the event {R > 1}, we can apply Lemma 2.2 for a = 1 and b = R , which together with the convexity yields Again, by stationarity of X and absolute continuity of μ, almost surely, we can further bound the right-hand side of (24) from above, which yields By assumption (6) we have E[R ] < ∞ and hence P(R < ∞) = 1. We can thus estimate for all α > 0 using Hölder's inequality, As above, the assumptions (5) and (6) now guarantee summability. This proves Proposition 1.2 for the number of edges and thus of Theorem 1.1 part (ii).

Poisson-Delaunay Tessellations: Proof of Theorem 1.1 Part (iii) and Proposition 1.2 part (i)
The case of the DT is the most difficult one to handle, essentially since in this case, existence of points close to the origin does not automatically eliminate the influence of other distant points. To keep the argument simple, we thus only treat the case here where the underlying point process is a homogeneous PPP. Recall the definition of W a from (18). Our first step towards the proof of Theorem 1.1 (iii) is to verify that there exists a fixed α > 0 such that E[exp(αW a )] < ∞ holds for any a > 0. Let us write X λ to indicate the intensity λ in the underlying PPP and write S λ D = S D (X λ ) and W λ a for the number of edges of S λ D intersecting with B a . In particular, choosing a = λ = 1, (10) follows from this proposition for the Poisson-DT for small α > 0, which proves Proposition 1.2 part (i) for the Poisson-DT. The proof rests on a comparison on the exponential scale.
Proof For x ∈ R d , let Q r (x) denote the box of side length r centered at x. We define the finest discretization of R 2 into boxes such that every box in the 2-annulus contains points. Note that R is almost surely finite. For k ∈ N such that k > 2a , Note that once k > 2a , the right-hand side of (27) does not depend on a. Since these terms are summable from k = 1 to ∞, P(R = 2a ) tends to one and thus E[R] tends to infinity as a → ∞.
In the event {R = k} for some k ≥ 2a, the points of ∂ Q 3k (o) are within a distance at most √ 2k from the centroid of their Voronoi cell. Among these Voronoi cells, the neighboring ones are separated by a Voronoi edge and hence their cell centroids are Delaunay neighbors. The Delaunay edges connecting the centroids of the successive cells yield a closed path in the Delaunay graph surrounding B a . This path defines a bounded region in which both endpoints of any Delaunay edge intersecting B a are located. Further, this region is fully contained Hence, since the restriction of the Delaunay triangulation is a planar graph, using Euler's formula we arrive at Q 6k (o)).

Now we can use Hölder's inequality, the Laplace transform of a Poisson random variable and (27) to estimate
But the right-hand side is finite for α < 1 6 log 1 + 1 72 for all a > 0 and λ > 0, as asserted.
We have the following corollary of Proposition 2.3 for the total edge length. Further, we have the following scaling relation for λ, r > 0: Indeed, since X λ , X λ/r 2 are homogeneous PPPs with intensities λ, λ/r 2 , respectively, we have that X λ/r 2 ∩ B r equals r (X λ ∩ B 1 ) in distribution. Thus, S λ/r 2 D ∩ B r is equal to a rescaled version of S λ D ∩ B 1 in distribution where the length of each edge is multiplied by r . This implies the statement (28).  (1)). Let r > 0 be sufficiently large such that α/r < α c (1). Note that α c (a) = 1 a α c (1). Further, observe that the value α c (1) is independent of the intensity parameter of the underlying PPP. These imply that for any λ > 0, we have Choosing λ = 1/r 2 implies (29) with a = r everywhere. This concludes the proof.

Line Tessellations: Proof of Theorem 1.1 Part (iv)
Proof We use the notation of Sect. 1. Since for any line it suffices to show that under the assumption (7) the number of lines of S L intersecting with B 1 has exponential moments up to β . Now, a line l i in R 2 intersects with B 1 if and only if its distance parameter X i,1 is at most one in absolute value, independently of its angle parameter X i,2 ∈ [0, 2π]. By the assumption (7), the number of such lines has exponential moments up to β .

Proof of Proposition 1.2 part (ii)
Note that any edge of the VT, DT or JMT that intersects with B 1 is adjacent to precisely two cells intersecting with B 1 , whereas if W = 0, then V = 1, and thus we have the trivial bound V ≤ 2W + 1. Thus, the assertion (11) for any given α/2 > 0 follows from the assertion (10) for the same α.

Nested Tessellations: Proof of Corollary 1.3 and Proposition 1.4
Proof of Corollary 1.3. We write S for a fixed tessellation process that equals S i , i ∈ J , in distribution, and we define V according to (12) For the first factor on the right-hand side, note that we can bound as an inequality in [0, ∞]. From this, (i) follows immediately. As for (ii), let us assume that M α < ∞ holds for some α > 0 and N β < ∞ holds for some β > 0. Then, the moment generating function R → [0, ∞], β → N β is continuous (in fact, infinitely many times differentiable) in an open neighborhood of 0, which implies that lim β→0 N β = N 0 = 1. Analogous arguments imply that lim α→0 log M α = 0. Hence, there exists α > 0 such that N log M α < ∞, which implies (ii).
. Now, the collection of cells of S o intersecting Q 1 is given as We write S i, j for the second-layer tessellation corresponding to S N in the cell C i, j and By Hölder's inequality, it suffices to verify the existence of all exponential moments for each of the three terms on the right-hand side separately. The first term has all exponential moments thanks to the assumption of Proposition 1.4. Further, by symmetry between the second and the third term, it suffices to show existence of all exponential moments for one of them; we will consider the second term.
Since for fixed i ∈ {1, . . . , ..,N v +1 are independent Poisson random variables with parameters summing up to λ v , it follows that their superpo- which is further equal to Since the right-hand side is finite, we conclude the proof of the proposition.

Palm Versions of Tessellations: Proof of Corollary 1.5
We handle each case separately.
Proof of Corollary 1.5 for the Poisson-VT Corollary 1.5 follows directly from Lemma 2.1 and the Slivnyak-Mecke theorem (see e.g. [11, Section 9.2]). Indeed, since Lemmas 2.1 uses no information about the distribution of X but only the definition of a Voronoi tessellation, these lemmas remain true after replacing S * by S. Next, the Palm version X * of the underlying PPP equals X ∪ {o} in distribution by the Slivnyak-Mecke theorem, in particular, it contains o almost surely. Thus, using the aforementioned versions of Lemma 2.1 (for a = b = 1), we deduce that |S V ∩ B 1 | is stochastically dominated by 2π(#(X ∩ B 4 ) + 1). This random variable has all exponential moments, hence the corollary.  (28) holds with S λ D replaced by its Palm version (S λ D ) * for all λ > 0, and then one can complete the proof of Corollary 1.5 for the Poisson-DT analogously to the final part of the proof of Theorem 1.1 (iii).

Proof of Corollary 1.5 for the Poisson-LT
As already mentioned in Sect. 1, S * equals S L (X * ) where X * = X ∪{(0, Φ)}, with Φ being a uniform random angle in [0, π) that is independent of X .Thus, S * = S∪{l}, where l = {x ∈ R 2 : x 1 cos Φ +x 2 sin Φ = 0}. Since the intersection of l with B 1 has length 2, the corollary in the case of a Poisson-LT follows directly from Theorem 1.1 part (iv). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Proof of Corollary 1.5 for the MG
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