The β -Delaunay tessellation II: the Gaussian limit tessellation *

We study the weak convergence of β - and β (cid:48) -Delaunay tessellations in R d − 1 that were introduced in part I of this paper, as β → ∞ . The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in R d − 1 × R . The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers’ equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it is used to deﬁne a new stationary random simplicial tessellation in R d − 1 . As for the β - and β (cid:48) -Delaunay tessellation, the distribution of volume-power weighted typical cells in the Gaussian-Delaunay tessellation is explicitly identiﬁed, establishing thereby a new bridge to Gaussian random simplices. Also major geometric characteristics of these cells such as volume moments, expected angle sums and also the cell intensities of the Gaussian-Delaunay tessellation are investigated.


Introduction
In part I of this paper [9] we introduced two new classes of stationary random simplicial tessellations in R d−1 , the β-Delaunay tessellation (with β > −1) and the β -Delaunay tessellation (with β > (d + 1)/2). Their construction, which we recall in detail in Section 3.2 below, is based on a paraboloid growth and hull processes, which were originally introduced in [3,28] in order to describe the asymptotic geometry of random convex hulls in the unit ball. The definition of the paraboloid growth and hull processes in turn is based on a space-time Poisson point process on the product space R d−1 × R + (for the β-Delaunay tessellation) or R d−1 × R * − := R d−1 × (−∞, 0) (for the β -Delaunay tessellation) whose intensity measure is translation invariant in the spatial (i.e. R d−1 ) component and has a power-law density function in the time or height (i.e. R + or R * − ) coordinate. We used these processes to generalize the well-known construction of the Poisson-Delaunay tessellation, which is one of the most classical tessellation models considered in stochastic geometry and related fields. Moreover, in [9] we gave an explicit description of weighted typical cells of both the βand the β -Delaunay tessellations from which we computed several key characteristic quantities such as volume moments, expected angle sums and also cell intensities. In particular, this demonstrates that both the βand β -Delaunay tessellations are analytically tractable, which makes them attractive alternatives to the classical Poisson-Delaunay tessellation. Let us point out that the latter are widely used in wireless network modelling [1,6,11] or material sciences [24].
In the present paper we study the limiting behaviour of the sequence of βand β -Delaunay tessellations in R d−1 , as β → ∞. Under a suitable rescaling, one can verify (see Lemma 4.1 below) that the Poisson point processes underlying the constructions of the paraboloid growth and hull processes (and hence the βand β -Delaunay tessellation) converge in distribution to a Poisson point process on R d−1 × R whose intensity measure is again translation invariant in the spatial component and has density e h/2 in the time or height coordinate. Such Poisson point processes together with their parabolic hulls have previously appeared in the literature, namely in the expectation and variance asymptotics for the number of shocks in the solution of the inviscid Burgers equation [2], as well as in the description of geometric properties of convex hulls of Gaussian random points [4]. A closely related construction appeared as an example of a maxstable process in [5,30]. We use this point process to construct the related paraboloid growth and hull process and to define another stationary random simplicial tessellation in R d−1 , which we call the Gaussian-Delaunay tessellation. Given these constructions, it seems natural to expect a similar (functional) convergence on the level of the random tessellations themselves, i.e., that, as β → ∞, both the βand β -Delaunay tessellation weakly converge to the Gaussian-Delaunay tessellation. Of course, in order to make this rigorous, one needs to make precise what weak convergence of random tessellations formally means. However, this is by far not straightforward, since there is no natural topological structure on the space of tessellations and it is not clear whether the space of tessellations of R d−1 is even Polish. This turns problems related to the convergence of random tessellations into a highly non-trivial task, which has only very rarely been tackled in the existing literature (see [20,21] for the only example we are aware of). One way to work around this problem is by identifying a tessellation with its skeleton, by which we mean the union of the boundaries of all of its cells. From here on one can work with the well-established notion of weak convergence of random closed sets with respect to the usual Fell topology, which is conveniently handled by means of the so-called capacity functional, see [15,Chapter 16] or [27,Chapter 2]. Eventually, we deal with the capacity functional of the βand β -Delaunay tessellation by resorting to the parabolic growth and hull processes underlying their constructions and by proving a kind of localization property for these processes, which is very much in the spirit of the well established geometric limit theory of stabilization for which we refer to the survey articles [29,33]. This constitutes the most technical part of this paper, in which we show that, after suitable rescaling, the capacity functional of the βand β -Delaunay EJP 27 (2022), paper 62. tessellation converges to that of the Gaussian-Delaunay tessellation, proving thereby the weak convergence of the random tessellations mentioned earlier. In addition, our proof shows convergence in a topology which is much stronger than the usual Fell topology on the space of closed subsets of R d−1 . This is of interest, because quantities like the number of cells or the total surface content in a bounded window become continuous functionals in this stronger topology, but are not continuous with respect to the Fell topology. This new perspective on the weak convergence of random tessellations (or more general random closed sets) is of independent interest and might be fruitful also in the study of functional limit theorems of other stochastic geometry models beyond the ones we investigate in this paper.
It is another goal of this paper to discuss and to determine principal geometric properties and characteristics of Gaussian-Delaunay tessellations. For that reason we follow the lines of part I [9], introduce the class of volume-power weighted typical cells and compute explicitly their volume moments, expected angle sums and cell intensities. In particular, this class of cells includes the typical cell (in the usual sense of Palm calculus) and, up to translation, the zero-cell (the almost surely uniquely determined cell containing the origin) of the Gaussian-Delaunay tessellation. The results we obtain in this context will be a consequence of a probabilistic description of volume-power weighted typical cells in terms of Gaussian random simplices. As for the βand the β -Delaunay tessellation this demonstrates that also the new Gaussian-Delaunay tessellation is an analytically tractable model of a simplicial random tessellation. In particular, the fact that the typical cell has the same distribution is a volume-weighted Gaussian random simplex is the reason why we call the limiting object the Gaussian-Delaunay tessellation.
The remaining parts of this paper are organized as follows. In Section 2 we gather some necessary background material, especially about point processes and tessellations (Section 2.2) and the construction of Laguerre diagrams and tessellations (Section 2.3). In Section 3 we explain the paraboloid growth and hull processes that lead to the definition of β-, β -and Gaussian-Delaunay tessellations. Our main result about the convergence of βand β -Delaunay tessellations towards the Gaussian-Delaunay tessellation is presented and proved in Section 4. In the final Section 5 a probabilistic representation of volume-power weighted typical cells is found and some key geometric characteristics of such cells are computed.

Notation
A centred unit Euclidean ball in R d is denoted by B d and its volume is given by . By B r (w) we denote the closed (d − 1)-dimensional ball of radius r > 0 centred at w ∈ R d−1 . We will also use a simplified notation for the ball B r := B r (0) centred at w = 0.
In what follows we will often use two equivalent systems of coordinates, namely, we represent points x ∈ R d as x := (v, h) ∈ R d with v ∈ R d−1 (usually called the spatial coordinate) and h ∈ R (referred to as the height or time coordinate). Finally, the Euclidean norm is indicated by · . Given a set C in a Euclidean space we denote by conv(C) its convex hull, by int(C) its interior and by ∂C its boundary. We denote by R + the set of all non-negative real numbers and by R * − := (−∞, 0) the set of all negative real numbers.

General point processes and tessellations
Let (X, X ) be a measurable space supplied with a σ-finite measure µ. By N(X) we denote the space of σ-finite counting measures on X. The σ-field N (X) is defined as the smallest σ-field on N(X) such that the evaluation mappings ξ → ξ(B), B ∈ X , ξ ∈ N(X), are measurable. A point process on X is a measurable mapping with values in N(X) defined over some fixed probability space (Ω, F, P). By a Poisson point process η on X with intensity measure µ we understand a point process with the following two properties: (i) for any B ∈ X the random variable η(B) is Poisson distributed with mean µ(B); (ii) for any n ∈ N and pairwise disjoint sets B 1 , . . . , B n ∈ X the random variables η(B 1 ), . . . , η(B n ) are independent.
We refer to [27,Chapter 3] for further background material on Poisson point processes.
Next, we introduce the notion of a general random tessellation in R d−1 and give a brief overview of some basic properties, which are used throughout the paper. For a more detailed discussion we refer to [27,Chapter 10]. A tessellation M in R d−1 is a locally finite countable system of compact, convex subsets of R d−1 having interior points, covering the space and having disjoint interiors. The elements of a tessellation M are called cells and every cell is a convex polytope. Given a polytope P ⊂ R d−1 denote by F k (P ) the set of its k-dimensional faces and let F(P ) := d−1 k=0 F k (P ) be the set of all faces of P . A tessellation M is called face-to-face if for all P 1 , P 2 ∈ M we have P 1 ∩ P 2 ∈ (F(P 1 ) ∩ F(P 2 )) ∪ {∅}.
A face-to-face tessellation in R d−1 is called normal if each k-dimensional face of the tessellation is contained in precisely d − k cells, k ∈ {0, 1, . . . , d − 2}. Further, we denote by M the set of all face-to-face tessellations in R d−1 . By a random tessellation in R d−1 we understand a particle process X in R d−1 (in the usual sense of stochastic geometry, see [27]) satisfying X ∈ M almost surely.

Laguerre tessellations
One of the most well studied type of tessellations is the classical Voronoi tessellation, see, for example, [22,31,27]. A Laguerre tessellation can be considered as a generalized (or weighted) version of a Voronoi tessellation and was intensively studied in [17,16,26]. In this subsection we only briefly recall some facts about Laguerre tessellations. For more details we refer the reader to part I of this paper [9, Section 3.2 -3.5].
The construction of a Laguerre diagram is based on the notion of a power function.
For v, w ∈ R d−1 and W ∈ R we define the power of w with respect to the pair (v, W ) as where W is referred to as the weight of the point v. Let X ⊂ R d−1 × R be a countable set of weighted points in R d−1 such that min (v,W )∈X pow(w, (v, W )) exists for each w ∈ R d−1 .
We emphasize that it is not necessarily the case that a Laguerre cell is non-empty or that it contains interior points. The collection of all Laguerre cells of X with non-vanishing topological interior is called the Laguerre diagram of X and we write It should be mentioned that a Laguerre diagram is not necessarily a tessellation, since the latter property strongly depends on the geometric properties of the point set X.
In part I of this paper we proved the following lemma, which gives a sufficient condition for a point process ξ in R d−1 × E, E ⊂ R a Borel set, which ensures that L(ξ) is in fact almost surely a random tessellation.
(P2) For every w ∈ R d−1 and every t ∈ E there are almost surely only finitely many Then L(ξ) is a random tessellation in R d−1 and, moreover, L(ξ) is normal with probability 1.

Paraboloid growth and paraboloid hull processes
In this Section 3.1 we introduce a paraboloid growth process with overlaps (or, simply, paraboloid growth process) and a paraboloid hull process. They represent a powerful tool which we will use for constructing β-, β -and Gaussian-Delaunay tessellations and investigating their properties. The paraboloid growth and hull processes were first introduced in [28,3] in order to study the asymptotic geometry of the Voronoi flower and the convex hull of Poisson point processes in the unit ball, respectively.
Let Π + and Π − be the upward and downward standard paraboloids We denote the apex of the paraboloid Π ±,x by apex Π ±,x := x. Further, given a set A ⊂ R d we put Following the definition from [3] for a given locally finite point set X ⊂ R d we introduce the paraboloid growth model Ψ(X): In particular, given a Poisson point process ξ, Ψ(ξ) is called the generalized paraboloid growth process with overlaps or simply the paraboloid growth process. In other words, Ψ(ξ) is a Boolean model (germ-grain model) in R d with paraboloid grains Π + , see [27,Chapter 9]. The name generalized paraboloid growth process with overlaps comes from the original interpretation of this construction [28]  We denote by ext(Ψ(ξ)) the set of all extreme points of the paraboloid growth process Ψ(ξ).
Further, we introduce the paraboloid hull process, which can be regarded as a dual to the paraboloid growth process and was considered in [3], see Figure 1 for an illustration. The paraboloid hull process is designed in order to exhibit properties analogous to those of convex polytopes, where paraboloids are playing a role of hyperplanes, spatial coordinates v are playing a role of spherical coordinates and hight coordinates h are playing a role of radial coordinates.
For any collection to be the intersection of aff(v 1 , . . . , v k ) × R with any translate of Π − containing all the points x 1 , . . . , x k . It should be mentioned here that although such translates of Π − are not unique for k < d, their intersections with aff(v 1 , . . . , v k ) × R all coincide so that Π(x 1 , . . . , x k ) is uniquely determined. Further we define the parabolic face Π[x 1 , . . . , Following [3] we say that a set A ⊂ R d is upwards paraboloid convex if for each  Finally, given a locally finite point set X ⊂ R d we define its paraboloid hull Φ(X) to be the smallest upwards paraboloid convex set containing X. Given a Poisson point process ξ, we define the paraboloid hull process as Φ(ξ). Following the arguments analogous to [3, Lemma 3.1] one has that, with probability one, where ξ d = is the collection of all d-tuples of distinct points of ξ, see Figure 2 for an illustration. We shall point out that with probability one any d distinct points (x 1 , . . . , x d ) ∈ ξ d = have affinely independent spatial coordinates and the right-hand side in (3.1) is almost surely well defined.
x d ] and Π[y 1 , . . . , y d ] are called co-paraboloid provided that Π(x 1 , . . . , x d ) = Π(y 1 , . . . , y d ). Note, that although all points in {x 1 , . . . , x d } and {y 1 , . . . , y d } must be distinct according to the definition of a paraboloid sub-facet, the sets {x 1 , . . . , x d } and {y 1 , . . . , y d } themselves do not need to be disjoint. The definition of co-paraboloid sub-facets is equivalent to saying that all points y 1 , . . . , y d , x 1 . . . , x d are lying on the same shift of standard downward paraboloid Π − . By a paraboloid facet (or (d − 1)-dimensional paraboloid face) of Φ(ξ) we understand the union of each maximal collection of co-paraboloid sub-facets. If ξ is a Poisson point process satisfying (P3) each paraboloid facet of Φ(ξ) with probability one consists of exactly one sub-facet. We can thus say that with probability one Π[x 1 , . . . , More formally, the vertex is a zero dimensional face of Φ(ξ), which arises as an intersection of suitable d-tuple of adjacent (d − 1)-dimensional faces of Φ(ξ). The collection of all vertices of Φ(ξ) is denoted by vert(Φ(ξ)) and it is easy to see that vert(Φ(ξ)) ⊂ ξ.
Using the paraboloid growth and paraboloid hull processes we can construct random diagrams in R d−1 which under suitable conditions on the underlying point process ξ give rise to random tessellations. Consider first the paraboloid growth process Ψ(ξ). Given a point x = (v, h) ∈ ext(Ψ(ξ)) define the Ψ-cell of x as  In other words w belongs to Thus, the Ψ-cell of an extreme point x of the paraboloid growth processes Ψ(ξ) has non-empty interior and coincides with the Laguerre cell C(x, ξ), which we defined in Section 2.3. Next, we construct the diagram M Ψ (ξ) as the collection of all Ψ-cells: We directly have that M Ψ (ξ) ⊂ L(ξ). On the other hand, if the Laguerre cell C(x, ξ) has non-empty interior then x ∈ ext(Ψ(ξ)) and thus M Ψ (ξ) = L(ξ). Now, consider the paraboloid hull processes Φ(ξ) and construct a diagram M Φ (ξ) in the following way: for any paraboloid facet f of Φ(ξ) with vertices x 1 , . . . , x m ∈ ξ a polytope conv(v 1 , . . . , v m ) belongs to M Φ (ξ). As already mentioned, if the point process ξ satisfies condition (P3), then with probability one any paraboloid facet of Φ(ξ) has exactly d vertices and the construction above can be alternatively described as follows: for any collection If the point process ξ satisfies conditions (P1) -(P3) of Lemma 2.1 then M Φ (ξ) and M Ψ (ξ) are a random tessellations in R d−1 and M Φ (ξ) = L * (ξ) almost surely.

β-Delaunay and β -Delaunay tessellations
The βand β -Delaunay tessellations have been introduced in part I [9] of this paper, where two alternative definitions using the concept of Laguerre tessellations and the notion of paraboloid hull processes were given. The first approach is more suitable for verifying properties of the tessellation and defining the typical cell, while the second approach is used for investigating the probabilistic behaviour of the typical cell. For convenience of the reader and to make the paper more self-contained will briefly recall both definitions here.

The Gaussian-Delaunay tessellation
Following the construction of βand β -Delaunay tessellations let us now define what we mean by the Gaussian-Delaunay tessellation. It is based on a Poisson point process ζ in R d−1 × R whose intensity measure has density As in the previous subsection we will consider two alternative definitions of the resulting tessellation. First of all, we need to ensure that properties (P1) -(P3) from Lemma 2.1 hold for the point process ζ. Property (P1) holds because the projection of the Poisson point process ζ to the space component R d−1 is with probability one a dense subset of R d−1 . Property (P3) also holds for any Poisson point process in R d whose intensity measure is absolutely continues with respect to the Lebesgue measure, see the proof of [9,Lemma 3.3]. In order to check (P2) it is enough to show that  we obtain Hence, the Laguerre diagram L(ζ) based on ζ is an almost surely normal random tessellation in R d−1 . The dual tessellations D := L * (ζ) is with probability one a simplicial random tessellation and is called the Gaussian-Delaunay tessellation in R d−1 . This terminology is motivated by the fact that the typical cell of the Gaussian-Delaunay tessellation is a (volume-weighted) Gaussian random simplex, see Remark 5.2 below.
Alternatively, based on the paraboloid hull processes Φ(ζ) one can construct the diagram M Φ (ζ) as described in Section 3.1 and, as before, conclude that with probability one M Φ (ζ) coincides with Gaussian-Delaunay tessellation L * (ζ) = D.
We emphasize at this point that the β-, β -and the Gaussian-Delaunay tessellation in R d−1 are in fact stationary random tessellations, meaning that their laws are invariant under shits in R d−1 . The reason for this is simply that the Poisson point processes η β , η β and ζ underlying the construction of these tessellations are stationary under shifts of their spatial components.

Convergence to the Gaussian limiting tessellation 4.1 Convergence of the point processes
Let us recall that the β-Delaunay and β -Delaunay tessellations were build on the Poisson point processes η β,γ in R d−1 × R + and η β,γ in R d−1 × R * − , whose intensity measures have densities with respect to the Lebesgue measure, respectively. For β > 0 consider the transforma- We are interested in the rescaled point processes where we choose the intensity γ as √ 2β. For any fixed C > 0 and sufficiently large β, these can be considered as point processes on R d−1 × [−C, C]. Unfortunately, it is not possible to consider them as point processes on R d−1 × R because the corresponding point configurations are not locally finite.
We will start by proving the following lemma, which shows that both ζ β and ζ β converge in distribution (with respect to the topology of vague convergence on the space of locally finite counting measures on R d−1 × [−C, C]) to the Gaussian limiting point process ζ, as β → ∞, which we defined in Section 3.3. Formally, this means that for any finite collection of relatively compact Borel sets B 1 , . . . , . . , n} one has that the n-dimensional random vector (ζ β (B 1 ), . . . , ζ β (B n )) converges in distribution to the n-dimensional random vector (ζ(B 1 ), . . . , ζ(B n )), as β → ∞ (and similarly with ζ β replaced by ζ β when the β -Delaunay tessellation is considered). We indicate this by writing ζ β d −→ ζ, as β → ∞. We refer to [15,Chapter 16] for background material on the convergence of point processes. Proof. We will only prove the result for ζ β , the convergence for ζ β can be shown in the same way. By the mapping property of Poisson point processes, ζ β is a Poisson point process on R d−1 × [−C, C], if β is sufficiently large, with intensity measure satisfying 2β v and s = 2β(h − 1) this transforms as follows: Since, by Stirling's formula, c d,β is asymptotically equivalent to (β/π) d/2 , as β → ∞, we conclude from the dominated convergence theorem that This is equivalent to say that

Convergence of the tessellations
Motivated by Lemma 4.1, it is natural to expect that a similar distributional convergence holds on the level of tessellations as well. However, it is not straightforward to introduce a suitable notion of weak convergence of random tessellations. One reason behind is that there is no natural topological structure on the space of tessellations and it is not clear whether the space of tessellations in R d−1 is even Polish or not (see the discussion in [10, Section 2.3]). A possible way out is to view a random tessellation as a random closed set in the common sense of stochastic geometry and to work with the weak convergence of random closed sets with respect to the so-called Fell topology on the space C, the space of all compact subsets of R d−1 , see [27,Chapter 2] for a general introduction to the theory of random closed sets. In our case this turns out to be possible since for any β > −1 (or β > (d + 1)/2) the β-(or β -)Delaunay tessellation and also their common limit, the Gaussian-Delaunay tessellation, are in fact tessellations of R d−1 .
To formalize this idea we need to introduce some notation. Given a stationary random Our principal goal in this section is to prove the following theorem, which is the counterpart to Lemma 4.1 on the level of tessellations. More precisely, we shall prove that, as β → ∞, the random closed sets D β and D β converge weakly (with respect to the Fell topology on the space C) to the random closed set D. Formally, this means that the capacity functionals T β and T β both converge, as β → ∞, to the capacity functional T of the Gaussian-Delaunay tessellation. For background material on the convergence of random sets we refer to [15,Chapter 16] or [27,Chapter 2] as well as to the explanations we give after the statement of our result.  Then the random closed sets √ 2βD β := skel( √ 2β D β ) and √ 2βD β := skel( √ 2β D β ) converge weakly to the random closed set D, as β → ∞. This follows from the fact that D β = √ 2βD β and D β = √ 2βD β , and is due to the property of the transformation Q β , which preserves the parabolic structure of the set. Namely the image of paraboloid for all C ∈ C such that T Z (int C) = T Z (C). The proof of Theorem 4.2 consists, roughly speaking, of three steps that can be summarized as follows: Step Step 2 (Lemma 4.5 in Section 4.3): In the second step we consider a kind of localization property in the spatial coordinate. More precisely, for any given R > 0 we construct some r > 0 such that the restricted random sets D β ∩ B R , D β ∩ B R and D ∩ B R are determined with high probability once the paraboloid hull processes underlying their construction are known in the cylinder B R+r × R.
Step 3 (Section 4.4): The last step consists of a coupling argument and is essentially based on the estimates we derived in Step 1 and Step 2. Using these results we obtain that with high probability the random sets D β ∩ B R , D β ∩ B R and D ∩B R are determined by the restrictions of corresponding Poisson point processes to some particular region in R d−1 × R. Thus, for any given ε > 0 and R > 0 we find a sufficiently large parameter β and construct a probability space such that with probability 1 − ε the random tessellations D β and D β restricted to B R coincide with the Gaussian limiting tessellation D within the same ball. In particular, this yields the convergence of the capacity functionals T β and T β of D β and D β , respectively, to the capacity functional T of the Gaussian-Delaunay tessellation.
Although the Fell topology on the space of closed subsets of R d−1 and the related weak convergence are widely used in the theory of random sets (see, for example, the monograph [19]), convergence in Fell topology does not imply convergence of many interesting quantities, such as the number of cells or faces within a given ball, the total surface area within a ball, and so on. However, the proof of Theorem 4.2 we give and, which we summarized above, actually shows weak convergence of D β and D β to D with respect to a much stronger topology on the space of closed subsets of R d−1 . To introduce this non-separable topology, let C ∈ C be a closed subset of R d−1 . The filter of neighbourhoods of this set is defined as follows. Take a radius R > 0 and consider the set O R (C) consisting of all closed subsets B ⊂ R d−1 such that the restrictions of B and C to the ball of radius R around the origin are equal, that is, B ∩ B R = C ∩ B R . One easily checks that the system O R (C) in fact defines a filter of neighbourhoods of C. A subset of C is called open if, together with each element C ∈ C it contains a set of the form O R (C) for some R > 0. This defines a (very strong) topology on C for which the geometric functionals mentioned above are continuous. However, we decided to present Theorem 4.2 for the Fell topology for simplicity and also since this is the much more classical notion used in the theory of random sets.

Technical preparations
The main idea of the proof of Theorem 4.2 is to use (4.2) and the connection between D β (and D β , D) and the boundaries of the paraboloid growth and hull processes described in Section 3.1. In this section we present a number of technical preparations for the proof of Theorem 4.2, which is deferred to the next section.
Lemma 4.4. Fix A > 0 and let T, t ∈ R. The following assertions hold.

(a) For any
Proof. In what follows, let us write ξ for one of the point processes ζ, ζ β or ζ β . We start by proving the first claim and we observe that by the definition  where we recall that for a particle x : Figure 6. This cannot hold for points (v , h ) with h ≤ T − 4A 2 and v ∈ B A , hence, since ξ is a Poisson point process. Next, we evaluate the last term for ξ = ζ, ξ = ζ β and ξ = ζ β . For ξ = ζ we have which together with (4.3) finishes the proof in this case. Analogously, from the definition of the intensity measure of the Poisson point processes ζ β for T > 4A 2 − 2β we get Applying inequality in [23, Paragraph 5.6.8] for quotients of gamma functions we conclude that Moreover since the function x → (1+x) 1/x is decreasing for x > −1 we get for β ≥ β 0 ≥ 1 and T ≥ 4A 2 that Finally, by definition of the intensity measure of the Poisson point processes ζ β and the fact that 1 − x < e −x for x < 1 we obtain for T < 4A 2 + 2β, As before, applying the inequality in [23, Paragraph 5.6.8] for quotients of gamma functions we get Substituting these estimates into (4.3) we finish the proof of the first claim. In order to prove the second claim we argue similarly and start by writing As before, given a particle x := (v , h ) ∈ ξ the condition that inf (v,h)∈Π+,x,v∈B A h > t for Consider the region which in R 2 and R 3 is illustrated in Figure 7. Then P inf and it remains to determine E[ξ ∩ R(A, t)] for ξ = ζ, ξ = ζ β and ξ = ζ β . We start by analysing the second case, where Fubini's theorem implies that where {x} stands for the fractional part of a real number x ∈ R. And since 1 + x ≤ e x for x > −1 we see that Applying the same arguments to the Poisson point process ζ β we obtain for t < 2β Next, for i ∈ {0, 1, . . . , d − 1} we compute the integral appearing in the last expression: Again, by inequality [23, Paragraph 5.6.8] for β ≥ β 0 > (d + 1)/2 we obtain , and, thus, Combining this with c d,β β −d/2 < π −d/2 and (4.8) we obtain With a similar computation one also shows that This completes the proof of the lemma.
In the next lemma we prove a kind of localization property for β-, β -and Gaussian-Delaunay tessellations. It is very much in the spirit of the geometric limit theory of stabilization for which we refer to the survey articles [29,33]. For R ≥ 1 and r > 0 we say that D β ∩ B R is determined by particles (v, h) ∈ ∂Ψ(ζ β ) with v ∈ B R+r , provided that the β-Delaunay tessellation within B R is unaffected by changes of the point configuration ζ β outside of B R+r × R. The same terminology is also applied if ζ β is replaced by one of the point processes ζ β or ζ.
3. for any ε ∈ (0, 1) and R ≥ 1 there existsr 0 :=r 0 (ε, R) > 0 such that for any r >r 0 we have By Lemma 4.4 and the law of total probability we have that According to the construction, D(ξ) coincides almost surely with the skeleton of the random tessellation M Φ (ξ), recall (3.2). Then D(ξ) ∩ B R is determined as soon as we know the location of all vertices of paraboloid facets of Φ(ξ) hitting the set B R × R and the event E(ξ) occurs if and only if there is a paraboloid facet of Φ(ξ) hitting the set B R × R and having a vertex    Figure 8.
In this case we obtain the following estimate: see Figure 9. Then For some a > 0 consider a partition of R d−1 into the boxes Q x,y , x ∈ aZ d−1 , y ∈ Z of the form where ⊕ denotes Minkowski addition. For fixed y ∈ Z, y ≥ 0, denote by Q x1,y , . . . , Q x m(y) ,y the m(y) boxes having non-empty intersection with the set Note that for y < 0 the set K(t, R, r, y) is empty, that is why we can restrict ourself to y ≥ 0.
Further, applying the Boole's inequality to (4.10) and using the fact that apexes of the parabolic facets of paraboloid hull process Φ(ξ) belong to the boundary of paraboloid growth process Ψ(ξ) (see also Figure 1) we see that h > t + r 2 /4 + y < ∞ y=0 m(y)p a (ξ, y). It remains to estimate m(y). Since all boxes intersecting K(t, R, r, y) for fixed y are included in the extended set 1≤i≤m(y) Q xi,y ⊂K(t, R, r, y) := [t + r 2 /4 + y, t + r 2 /4 + y we can conclude, that, Substituting this estimate into (4.12) we obtain 1 p a (ξ, y).
Let us point out that for r > √ 2d ≥ 2 all functions y → S i (y), 0 ≤ i ≤ d − 1 are strictly decreasing in y ≥ 0 and the sum can be estimated as The last integral is the incomplete gamma function Γ(i/2 + 1, r 2 /8) and its asymptotics, as r → ∞, is well-known. Using the recurrence relation Combining this with (4.13) we obtain, that In a next step, we observe that by estimate 2(a) of Lemma 4.4 for t = 2 ln ln 1 1 − ε/2 − 2(d − 1) ln(R + r) − d ln(2d + 4) + ln π and for R ≥ 1 we have that It is also easy to ensure, that for sufficiently big r (depending on R, ε and d) we have that r 2 > −4t + 16(d − 1)a 2 . Thus, substituting expression above into (4.14) we obtain for some constant c 1 (d) > 0, that Since lim r→∞ (R + r) 3(d−1) e −r 2 /8 = 0 and function on the right hand side of the last inequality is strictly decreasing for sufficiently large r, we may now choose r 0 (R, ε) > 0, depending on R, ε and d only in such a way that P 1 (ζ β ) ≤ ε/2 for any r > r 0 (R, ε).

Proof of the Theorem 4.2
We will give the proof only for D β . The proof for D β works in the same way. To this end, let C ∈ C. From the construction of the Gaussian-Delaunay tessellation D it is clear that the capacity functional of D satisfies T (C) = T (int C). Since C is bounded there exists R ≥ 1 such that C ⊂ B R . To simplify our notation, we denote the capacity functional of D β by T β ( · ) and that of the limiting random closed set D by T ( · ). By the law of total probability we have Further, according to Lemma 4.5 for any ε ∈ (0, 1) and r > max(r 0 (ε, R),r 0 (ε, R)) we have P( D β ∩ B R = D ∩ B R ) ≤ P(∂Ψ(ζ β ) = ∂Ψ(ζ) in B R+r × R) + 2ε, and from Lemma 4.4 1(a) and 1(c) for any T > T 0 (ε, R + r) we obtain However, the last probability is equal to total variation distance between the Poisson point processes ζ and ζ β (see [32,Equation (5.12)]), which according to [ Due to (4.4) and the fact that 1 + x ≤ e x for all x > −1 we obtain Then, since lim β→∞ c d,β β −d/2 = π −d/2 we conclude that there exists β 1 := β 1 (R, ε) such that for any β > β 1 we have I 1 (β) < ε. On the other hand, by (4.7) we obtain By (4.18) the function f is integrable and, hence, by dominated convergence theorem there exists β 2 := β 2 (R, ε) such that for any β > β 2 we have I 2 (β) < ε. This implies that Together with (4.17) this shows that, as β → ∞, for any C ∈ C the value of the capacity functional T β (C) of D β converges to T (C), the value of the capacity functional of D.
According to (4.2) this proves the theorem and, at the same time, the stronger version described at the end of Section 4.2.

Definition and probabilistic representation
In this section we investigate the distribution of typical cells in the Gaussian-Delaunay tessellation D. Typical cells of βand β -Delaunay tessellations have been considered in the first part of the article [9]. We have already introduced there the concept of weighted typical cells in a more general context, which we briefly recall for convenience. In this section we let ζ be the Poisson point processes in Let ζ * be defined as in (2.1) or, in other words, we can consider ζ * as the set of apexes of paraboloid facets of the paraboloid hull process Φ(ζ): Consider the random marked point process , whose marks are the associated and suitably centred Laguerre cells. For a given parameter ν ∈ R we now define a probability measure P ζ,ν on C , the space of non-empty compact subsets of R d−1 , by Note that λ ζ,ν might be infinite for some values of ν. The proof of Theorem 5.1 will show that 0 < λ ζ,ν < ∞, provided that ν ≥ −1. For such ν, a random simplex Z ν with distribution P ν := P ζ,ν is called the Vol ν -weighted (or just ν-weighted) typical cell of the Gaussian-Delaunay tessellation D. The following two special cases are of particular interest: (i) Z 0 coincides with the classical typical cell of D; (ii) Z 1 coincides with the volume-weighted typical cell of D, which has the same distribution as the almost surely uniquely determined cell containing the origin, up to translation.
The next theorem yields an explicit description of the distribution of the ν-weighted typical cell of the Gaussian-Delaunay tessellation. .

Remark 5.2.
The previous theorem yields the following convenient probabilistic description of the ν-weighted typical cell Z ν of the Gaussian-Delaunay tessellation D. Let Y 1 , . . . , Y d be d random vectors in R d−1 whose joint density is proportional to Then the random simplex conv(Y 1 , . . . , Y d ) has the same distribution as Z ν . In other words, the ν-weighted typical cell Z ν of D is a weighted Gaussian random simplex, where the weight is given by the (ν + 1)st volume power. As already explained in the introduction, this is the reason behind our terminology for the random tessellation D.

Remark 5.3.
It should be mentioned that the distribution P ν of the ν-weighted typical cell Z ν of Gaussian-Delaunay tessellation is independent of the intensity parameter γ of the underlying Poisson process ζ. Indeed, changing γ amounts to shifting the Poisson point process ζ in the vertical direction, which does not change the tessellation.
Proof of Theorem 5.1. We start by observing that, up to a multiplicative constant, P ν (A) is the same as Let (w, r) ∈ R d−1 × R denotes the apex of the paraboloid Π ↓ (x 1 , . . . , x d ). Then w = z(x 1 , . . . , x d ), and we can write v i = w + z i for i = 1, . . . , d with uniquely determined z 1 , . . . , z d ∈ R d−1 . Moreover, we have the relation h i = r − z i 2 for each i = 1, . . . , d. This allows us to introduce the transformation T : It is not hard to check that for the absolute value J(T ) of the Jacobian of T we have In fact, using the convention that |M | denotes the absolute value of the determinant of a matrix M , we have that Using the fact that ζ is a Poisson point process and inserting the precise form of its intensity measure, the probability in the last expression can be computed as follows: The remaining integral is determined by introducing spherical coordinates in R d−1 , which yields Putting everything together and simplifying the constant leads to Thus, using the abbreviation m d := √ 2γ √ π , we find that Now, we have to observe that the integral is the (ν + 1)st moment of the volume of a (d − 1)-dimensional Gaussian simplex, up to a normalizing constant. The value of the moment can be found in [18,Equation (70)] where ν is required to be integer (see also [8,Theorem 2.3 (a)], an extension to non-integer ν can be done as in the proof of Proposition 2.8 in [13]) and is given by . (5.4) Thus, .
As a consequence, simplification of the constants shows that

Expected angle sums, volume moments and cell intensities
The probabilistic representation of weighted typical cells in Theorem 5.1 allows us to determine various geometric quantities associated with such cells. We start with the expected angle sums of Z ν . For a simplex T = conv(X 1 , . . . , X d ) ⊂ R d−1 we denote by σ k (T ) := 1≤i 1 <...<i k ≤d F :=conv(X i 1 ,...,X i k ) β(F, T ) the sum of internal angles of T at all its k-vertex faces F , see [27,Chapter 6.5]. Further, by Σ d−1 we denote a (d − 1)-dimensional regular simplex. We recall that in [9, Proposition 6.3] of part I of this paper we have seen that, as β → ∞, the expected angle sums of the ν-weighted typical cell of a β-Delaunay tessellation approach those of a regular simplex. Given Theorem 4.2 it can now be anticipated that the expected angle sums of ν-weighted typical cell of the limiting Gaussian-Delaunay tessellation coincide with that of a regular simplex. Our next result, which is essentially a consequence of Theorem 5.1, confirms that this is indeed the case.
Remark 5.6. Note that [14,Remark 4.2], which has been used in the previous proof, requires ν to be integer but it is natural to conjecture that it remains true without this assumption. This in turn would imply that Corollary 5.5 also holds under the same circumstances.
Next, we compute the moments of the volume of Z ν . .
Plugging in the value for α d,ν completes the proof.
From the previous corollary we can derive a probabilistic representation of the volume of Z ν , which is similar to that in [8, Theorem 2.5(a)] for the volume of a Gaussian random simplex. To this end we recall that a random variable X k is said to have a χ 2 -distribution with κ > 0 degrees of freedom, provided it has density t → 1 2   ) . (5.6) Comparison of (5.5) and (5.6) proves the claim.
Corollary 5.7 also allows us to compute the intensity γ j (D) of j-dimensional faces of D, that is, γ j (D) is the mean number of j-dimensional faces of D per unit volume in R d−1 ; see [27, p. 450 and § 4.1] for an exact definition.