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STIT Tessellations have trivial tail \sigma-algebra

We consider homogeneous STIT tessellations Y in the \ell-dimensional Euclidean space and show the triviality of the tail \sigma-algebra. This is a sharpening of the mixing result by Lachi\`eze-Rey.


Introduction
Let Y = (Y t : t > 0) be the STIT tessellation process, which is a Markov process taking values in the space of tessellations of the ℓ-dimensional Euclidean space R ℓ . The process Y is spatially stationary (that is its law is invariant under translations of the space) and on every polytope with nonempty interior W (called a window) the induced tessellation process, denoted by Y ∧W = (Y t ∧W : t > 0), is a pure jump process. The process Y was firstly constructed in [12] and in Section 2 we give a brief description of it and recall some of its main properties.
In stochastic geometry, ergodic and mixing properties as well as weak dependencies in space are studied. E.g., Heinrich et al. considered mixing properties for Voronoi and some other tessellations, Poisson cluster processes and germgrain models and derived Laws of Large Numbers and Central Limit Theorems, see [3,6,5].
For STIT tessellations, Lachièze-Rey [8] showed that they are mixing. Schreiber and Thäle [15,16,17] proved some limit theorems. They provided Central Limit Theorems for the number of vertices and the total edge length in the twodimensional case (ℓ = 2). Furthermore, they proved that in dimensions ℓ ≥ 3 there appear non-normal limits, e.g. for the total surface area of the cells of STIT tessellations.
An issue is the problem of triviality of the tail σ-algebras for certain class of distributions. For random measures and point processes a key reference is the book by Daley and Vere-Jones ( [2], pp. 207-209).
In Section 3 of the present paper we introduce a definition for the tail σalgebra B −∞ (T) on the space T of tessellations in the ℓ-dimensional Euclidean space R ℓ . This definition relies essentially on the definition of the Borel σalgebra with respect to the Fell topology on the set of closed subsets of R ℓ (cf. [14]) as well as on the mentioned definition of the tail σ-algebra for random measures and point processes, as given in [2].
Our main result is formulated in in Section 4, Theorem 2: For the distribution of a STIT tessellation, the tail σ-algebra B −∞ (T) is trivial, i.e. all its elements (the terminal events) have either probability 1 or 0. A detailed proof is given in Section 5.
Finally, we compare STIT tessellations with Poisson hyperplane tessellations (PHT), and we show, that the tail σ-algebra B −∞ (T) is not trivial with respect to the distribution of PHT.

The STIT model 2.1 A construction of STIT tessellations
Let R ℓ be the ℓ-dimensional Euclidean space, denote by T the space of tessellations of this space as defined in [14] (Ch. 10, Random Mosaics). A tessellation can be considered as a set T of polytopes (the cells) with disjoint interiors and covering the Euclidean space, as well as a closed subset ∂T which is the union of the cell boundaries. There is an obvious one-to-one relation between both ways of description of a tessellation, and their measurable structures can be related appropriately, see [14,9].
A compact convex polytope W with non-empty interior in R ℓ , is called a window. We can consider tessellations of W and denote the set of all those tessellations by T ∧ W . If T ∈ T we denote by T ∧ W the induced tessellation on W . Its boundary is defined by ∂(T ∧ W ) = (∂T ∩ W ) ∪ ∂W .
In the present paper we will refer to the construction given in [12] in all detail (for an alternative but equivalent construction see [9]). On every window W there exists Y ∧ W = (Y t ∧ W : t > 0) a STIT tessellation process, it turns out to be a pure jump Markov process and hence has the strong Markov property (see [1], Proposition 15.25). Each marginal Y t ∧ W takes values in T ∧ W . Furthermore, for any t > 0 the law of Y t is consistent with respect to the windows, that is if W ′ and W are windows such that W ′ ⊆ W , then where ∼ denotes the identity of distributions (for a proof see [12]). This yields the existence of a STIT tessellation Y t of R ℓ such that for all windows W the law of Y t ∧ W coincides with the law of the construction in the window. A global construction for a STIT process was provided in [11]. A STIT tessellation process Y = (Y t : t > 0) is a Markov process and each marginal Y t takes values in T.
Here, let us recall roughly the construction of Y ∧ W done in [12].
Let Λ be a (non-zero) measure on the space of hyperplanes H in R ℓ . It is assumed that Λ is translation invariant and possesses the following locally finiteness property: (Notation ⊂ means strict inclusion and ⊆ inclusion or equality). It is further assumed that the support set of Λ is such that there is no line in R ℓ with the property that all hyperplanes of the support are parallel to it (in order to obtain a.s. bounded cells in the constructed tessellation, cf. [14], Theorem 10.3.2, which can also be applied to STIT tessellations).
the normalized probability measure. Let us take two independent families of independent random variables D = (d n,m : n, m ∈ N) and τ = (τ n,m : n, m ∈ N), where each d n,m has distribution Λ W and each τ n,m is exponentially distributed with parameter Λ([W ]).
• Even if for t = 0 the STIT tessellation Y 0 is not defined in R ℓ , we define Y 0 ∧ W = {W } the trivial tessellation for the window W . Its unique cell is denoted by C 1 = W .
• Any extant cell has a random lifetime, and at the end of its lifetime it is divided by a random hyperplane. The lifetime of W = C 1 is τ 1,1 , and at that time it is divided by d 1,1 into two cells denoted by C 2 and C 3 .
• Now, for any cell C i which is generated in the course of the construction, the random sequences (d i,m : m ∈ N) and τ = (τ i,m : m ∈ N) are used, and the following rejection method is applied: • When the time τ i,1 is elapsed, the random hyperplane d i,1 is thrown onto the window W . If it does not intersect C i then this hyperplane is rejected, and we continue until a number z i , which is the first index j for which a hyperplane d i,j intersects and thus divides C i into two cells C l1(i) , C l2(i) , that are called the successors of C i . Note that this random number z i is finite a.s. Hence the lifetime of C i is a sum τ * (C i ) := zi m=1 τ i,m . It is easy to check, see [12], that τ * (C i ) is exponentially distributed with parameter Λ([C i ]). Note that z 1 = 1 and τ * (C 1 ) = τ 1,1 .
• This procedure is performed for any extant cell independently. It starts in that moment when the cell is born by division of a larger (the predecessor) cell. In order to guarantee independence of the division processes for the individual cells, the successors of C i get indexes l 1 (i), l 2 (i) in N that are different, and that can be chosen as the smallest numbers which were not yet used before for other cells.
• For each cell C i we denote by η = η(i) the number of its ancestors and by (k 1 (i), k 2 (i), . . . , k η (i)) the sequence of indexes of the ancestors of and it dies at time κ(C i ) = κ(C i ) + τ * (C i ); for C 1 this is κ(C 1 ) = 0 and κ(C 1 ) = τ * (C 1 ). It is useful to put k η+1 (i) = i.
With this notation at each time t > 0 the tessellation Y t ∧ W is constituted by the cells C i for which κ(C i ) ≤ t and κ(C i ) > t. It is easy to see that at any time a.s. at most one cell dies and so a.s. at most only two cells are born. Now we describe the generated cells as intersections of half-spaces. First note that by translation invariance follows Λ([{0}]) = 0. Hence, all the random hyperplanes (d n,m : n, m ∈ N) a.s. do not contain the point 0. Now, for a hyperplane H ∈ H such that 0 ∈ H we denote by H + and H − the closed half-spaces generated by H with the convention 0 ∈ int(H + ). Hence, where the sign in the upper index determines on which side of the dividing hyperplane the cell C k l+1 (i) is located. Then any cell can be represented as an intersection of W with half-spaces, (2) In above relation we define the sign s(j, m) ∈ {+, −} by the relation C j ⊂ d s(j,m) j,m , for j, m ∈ N. Notice, that the origin 0 ∈ C i if and only if all signs in the previous formula (2) satisfy s(k l (i), m) = + and s(i, m) = +.
Obviously the set of cells can be organized as a dyadic tree, by the relation "C ′ is a successor of C". This method of construction is done in [12]. For the following it is important to observe that all the rejected hyperplanes d k l (i),m are also included in this intersection because the intersection with the appropriate half-spaces does not alter the cell. Although the third set zi−1 m=1 d in the intersection (2) does not modify the resulting set, we also include it because we will use this representation later.
In [12] it was shown that there is no explosion, so at each time t > 0 the number of cells of Y t ∧ W , denoted by ξ t , is finite a.s. Renumbering the cells, we write {C i t : i = 1, ..., ξ t } for the set of cells of Y t ∧ W .

Independent increments relation
The name STIT is an abbreviation for "stochastic stability under the operation of iteration of tessellations". Closely related to that stability is a certain independence of increments of the STIT process in time.
In order to explain the operation of iteration, for T ∈ T we number its cells in the following way. Assign to each cell a reference point in its interior (e.g. the Steiner point, see [14], p. 613, or another point that is a.s. uniquely defined). Order the set of the reference points of all cells of T by its distance from the origin. For random homogeneous tessellations this order is a.s. unique. Then number the cells of T according to this order, starting with number 1 for the cell which contains the origin. Thus we write C(T ) 1 , C(T ) 2 , . . . for the cells of T .
For T ∈ T and R = (R m : m ∈ N) ∈ T N , we define the tessellation T ⊞ R, referred to as the iteration of T and R, by its set of cells So, we restrict R k to the cell C(T ) k , and this is done for all k = 1, . . .. The same definition holds when the tessellation and the sequence of tessellations are restricted to some window.
To state the independence relation of the increments of the Markov process Y of STIT tessellations, we fix a copy of the random process Y and let . Then, from the construction and from the consistency property of Y it is straightforward to see that the following property holds This relation was firstly stated in Lemma 2 in [12].
Here the multiplication with 2 stands for the transformation

The space of tessellations and the tail σ-algebra
Let C be the set of all compact subsets of R ℓ . We endow T with the Borel σ-algebra B(T) of the Fell topology (also known as the topology of closed convergence), namely (As usual, for a class of sets I we denote by σ(I) the smallest σ−algebra containing I). Let us fix P a probability measure on (T, B(T)). All the sets are determined mod P, that is up to a P−negligible set. So, for E, D ∈ B(T) we For a window W we introduce Note that the set T ∧ W can be endowed with the σ−field Denoting the law of the STIT process Y by P it holds To avoid overburden notation and since there will be no confusion, instead of E ∧ W in the last formula we will only put E, so it reads Let (W n : n ∈ N) be an increasing sequence of windows such that We have In order to study the tail σ-algebra, we will also consider sets of tessellations which are determined by their behavior outside a window W , i.e. in its complement W c . We define the σ-algebra Let (W n : n ∈ N) and (W ′ n : n ∈ N) be a pair of increasing sequences of windows satisfying the conditions in (6).
. This equality allows us to define, in analogy with the definition done for point processes (see [2], Definition 12.3.IV), the tail σ-algebra on the space of tessellations.
is an increasing sequence of windows such that for all n ∈ N, W n ⊂ int W n+1 , and R ℓ = n∈N W n .
Note that (W n = [−n, n] ℓ : n ∈ N) satisfies (6), so it can be used in the above definition and also in the rest of the paper.
be an increasing sequence of windows satisfying (6). Let ǫ > 0 be fixed. Since D ∈ B(T), from (7) we get, From hypothesis (8) there exists a window W , depending on Since this occurs for all ǫ > 0 we conclude P(D) = P(D)P(D), so P(D) = 1 or 0.

Let us discuss what happens when
Note that if E ∈ I(T), D ∈ B(T) and E = D mod P, then D ∈ I(T). It is easily checked that I(T) ⊆ B(T) is a sub-σ-algebra, the invariant σ-algebra (see e.g. [2]). We have the inclusion relation, We note that (11) corresponds to propositions in [2] (pp. 206-210) for random measures. For completeness we will prove (11). We denote We have k > 1. Take N = k 2 and use k 2 > 2k to get ) and D m decreases with m . We conclude that D = m>1 D m satisfies P(E∆D) = 0 and D ∈ B −∞ (T). Hence, relation (11) holds.
So, when the tail σ-algebra B −∞ (T) is P−trivial, then P is ergodic with respect to translations, because every invariant set E ∈ I(T) also belongs to B −∞ (T) and so P(E) = 0 or P(E) = 1. We also have that if P is translation invariant and B −∞ (T) is P−trivial, then the action of translations is mixing. That is, This result is shown for random measures in Proposition 12.3.V. in [2]. But for completeness let us prove it. Let D, E ∈ B(T) and fix ǫ > 0. From (7) follows the existence of k and Our choice W n = [−n, n] ℓ implies that for all h ∈ R ℓ , N ∈ N with N > k and ). (13) The Decreasing Martingale Theorem (see e.g. [13]) yields Then for |h| > (N + k) √ ℓ we can use (13) to obtain, We conclude P(D ∩ (E + h)) → P(D)P(E) as h → ∞, so mixing is shown.

Main results
As already defined, Y = (Y t : t > 0) denotes a STIT tessellation process, defined by the measure Λ on H in R ℓ which satisfies the properties described in Section 2.1.
We say that at time t > 0 the tail σ-algebra B −∞ (T) is trivial for the STIT process Y if P t is trivial. Theorem 1 implies that P t satisfies the sufficient condition (8), which by Lemma 1 implies the triviality of the tail σ-algebra B −∞ (T). Hence, the following result holds.
Theorem 2 For all t > 0 the tail σ-algebra is trivial for the STIT process Y , that is ∀ E ∈ B −∞ (T) we have P(Y t ∈ E) = 0 or P(Y t ∈ E) = 1.
Relation (12) ensures that Theorem 2 is stronger than the mixing property shown by Lachièze-Rey [8].

Proof of Theorem 1
Let W ′ be a window. Since the measure Λ is supposed to be translation invariant, without loss of generality it can be assumed that the origin 0 ∈ int(W ′ ).
Let W be a window such that W ′ ⊂ int(W ). A key idea is the investigation of the probability that the window W ′ and the complement W c are separated by the STIT process Y ∧ W (we say W ′ encapsulated within W ) and thus the construction inside W ′ and outside W , respectively, are approximately independent.
For simplicity, denote by C t = C 1 t the (a.s. uniquely determined) cell of tessellation Y t ∧ W that contains the origin in its interior. Obviously, C 0 = W . Note that C t decreases as time t increases. On the other hand since W ′ ⊂ W , when we consider the STIT on W ′ we can take (Y t ∧ W ) ∧ W ′ .
Definition 2 Let be W ′ , W be two windows with 0 ∈ W ′ ⊂ int(W ) and let t > 0. We say that W ′ is encapsulated inside W at time t if the cell C t that contains 0 in Y t ∧ W is such that, We We denote the encapsulation time by That is, where the boundaries ∂W ′ and ∂W are completely separated by facets of the 0-cell before the smaller window W ′ is hit for the first time by a facet of the STIT tessellation.
We have {S(W ′ , W ) ≤ t} ∈ σ(Y s : s ≤ t), so it is a stopping time. Hence the variable S(W ′ , W ) is also a stopping time for the processes Y ∧ W . On the other hand notice that the distribution of S(W ′ , W ) does neither depend on the method of construction of the STIT process Y , nor on the window W where the construction is performed as long as W ⊆ W . In several proofs of the results we will assume that the starting process is Y , but in some others ones we will start from the STIT process Y ∧ W , as it occurs in Lemma 2.
Note that even if in the STIT construction we have "independence after separation", it has to be considered that the tessellation outside W also depends on the process until the separation time.
For two Borel sets A, B ⊂ R ℓ we denote by the set of all hyperplanes that separate A and B. This set is a Borel set in H.
For a window W (which is defined to be a convex polytope) we denote by {f W a : a = 1, . . . , q} the (ℓ − 1)-dimensional facets of W . Let W ′ be another window such that W ′ ⊂ int(W ). We denote by the set of hyperplanes that separate W ′ from the facet f W a of W . Note that all these sets are non-empty, and they are not necessarily pairwise disjoint.
There exists a finite family {G ′ a : a = 1, . . . , q} of pairwise disjoint nonempty measurable sets that satisfy (E.g. we can choose G ′ a = G a \ b<a G b , or we can partition q a=1 G a alternatively.) Lemma 2 Let W ′ , W be two compact convex polytopes, with W ′ ⊂ int(W ). Let {G ′ a : a = 1, . . . , q} be a finite class of nonempty disjoint measurable sets satisfying (15) and such that Λ(G ′ a ) > 0 for all a = 1, . . . , q. Then, Proof 2 Our starting point in the proof is the construction of the process Y ∧W . Because we assume 0 ∈ W ′ we focus on the genesis of the 0-cell C t = C 1 t , t ≥ 0, only. We use the representation (2), and we emphasize that in this intersection also participate the positive half-spaces of those hyperplanes d i,m which are rejected in the construction. . This point process can be considered as a marked hyperplane process where the marks are birth-times (or as a "rain of hyperplanes"). This choice of the intensity measure corresponds to the families D and τ of random variables in Subsection 2.1: the interval between two sequential births of hyperplanes is exponentially distributed with parameter Λ([W ]), and the law of the born hyperplanes is Λ W . Thus the S m are sums of i.i.d. exponentially distributed random variables which are independent of the d m . This corresponds to one of the standard methods to construct (marked) Poisson point processes (cf. e.g. [7]). Note that this Poisson process is used for the construction of the 0-cell exclusively.
Let η denote the number of ancestors of C t with indexes k 1 , . . . , k η , and Z k l = l m=1 z k l , l = 1, . . . , η + 1. Thus we can write formula (2) for the 0-cell as Define the random times These are the first times that a hyperplane of Φ falls into the respective sets. Note that these minima exist and are greater than 0 because all Λ [W ] (G ′ a ) > 0, and we are working on a bounded window W and Λ is assumed to be locally finite, so Λ W is a probability measure. Let On the other hand, if σ ′ ≥ M then W ′ is not intersected until the time M = max{σ a : a = 1, . . . , q}, so we have W ′ ⊆ C M . Then We have shown This relation implies straightforwardly the inclusion, Indeed, from M ≤ σ ′ we get S(W ′ , W ) ≤ M, and we use M ≤ t to get relation (19). We deduce Now, the sets [W ′ ], G ′ a , a = 1, . . . , q, are pairwise disjoint and therefore the restricted Poisson point processes × [0, ∞)), a = 1, . . . , q, are independent. Hence σ ′ , σ a , a = 1, . . . , q, are independent random variables. Then σ ′ and M are independent and we get Since σ ′ , σ a , a = 1, . . . , q, are exponentially distributed with the respective parameters Λ([W ′ ]), Λ(G ′ a ) > 0, a = 1, . . . , q, we find Now, by denoting the density functions of σ ′ by p ′ and those ones of σ i by p i respectively, we get Therefore, formula (16) follows.
Remark 1 (i) We emphasize that M ≤ min{σ ′ , t} is sufficient but not necessary for S(W ′ , W ) ≤ t. There can also be other ways to encapsulate W ′ within W than separating the complete facets of W by single hyperplanes. Alternative geometric constructions are possible.
(ii) It is well known in convex geometry, that [W ′ |f W a ] = ∅ for all a = 1, . . . , q. But depending on the support of the measure Λ (in particular, if Λ is concentrated on a set of hyperplanes with only finitely many directions) there can be windows W such that Λ([W ′ |f W a ]) = 0 for some a. In such cases the bound given in Lemma 2 is useless. Therefore, in the following, W will be adapted to Λ in order to have all Λ(G ′ a ) > 0. But here we will not try to find an optimal W in the sense that the quantities Λ(G ′ a ) could be somehow maximized.
and hence for Λ ⊥ we have the equality sign in (16).
We will use the following parameterization of hyperplanes. Let S ℓ−1 be the unit hypersphere in R ℓ . For H ∈ H, d(h) ∈ R denotes its signed distance from the origin and u(h) ∈ S ℓ−1 is its normal direction. We denote by H(u, d) the hyperplane with the respective parameters (u, d) ∈ R × S ℓ−1 . The image of the non-zero, locally finite and translation invariant measure Λ with respect to this parameterization can be written as the product measure where γ > 0 is a constant, λ is the Lebesgue measure on R and θ is an even probability measure on S ℓ−1 (cf, e.g. [14], Theorem 4.4.1 and Theorem 13.2.12).
Here θ is even means θ(A) = θ(−A) for all Borel sets A ⊆ S ℓ−1 . The property that there is no line in R ℓ such that all hyperplanes of the support of Λ are parallel to it, is equivalent to the property that θ is not concentrated on a great subsphere of S ℓ−1 , i.e there is no one-dimensional subspace L 1 of R ℓ (with the orthogonal complement L ⊥ 1 ) such that the support of θ equals G = L ⊥ 1 ∩ S ℓ−1 . Recall that W ′ is a window with 0 ∈ int(W ′ ).
Lemma 3 There exists a compact convex polytope W with facets f W 1 , . . . , f W 2ℓ and W ′ ⊂ int(W ), and pairwise disjoint sets Proof 3 For u ∈ S ℓ−1 we denote by H W ′ (u) the supporting (i.e. tangential) hyperplane to W ′ with normal direction u ∈ S ℓ−1 . By h W ′ (u) we denote the distance from the origin to H W ′ (u). This is the support function of W ′ , h W ′ (u) = max{< x, u >: x ∈ W ′ } (see [14], p. 600). With this notation we have H W ′ (u) = H(u, h W ′ (u)). Note that for d ∈ R the hyperplane H(u, d) is parallel to H W ′ (u) at signed distance d from the origin.
The shape of the window W will depend on Λ. We use some ideas of the proof of Theorem 10.3.2 in [14]. Under the given assumptions on the support of Λ there exist points u 1 , . . . , u 2ℓ ∈ S ℓ−1 which all belong to the support of θ and 0 ∈ int(conv{u 1 , . . . , u 2ℓ }), i.e. the origin is in the interior of the convex hull. Now, the facets f W a of W are chosen to have normals u a , and their distance from the origin is h W ′ (u a ) + 3, a = 1, . . . , 2ℓ. Formally, Notice that the described condition on the choice of the directions u 1 , . . . , u 2ℓ guarantees that W is bounded (see the proof of Theorem 10.3.2 in [14]).
The definition of the support of a measure and some continuity arguments (applied to sets of hyperplanes) yield that for all u a there are pairwise disjoint neighborhoods U a ⊂ S ℓ−1 such that θ(U a ) > 0, and for the sets of hyperplanes > 0 for all a = 1, . . . , 2ℓ. Since the U a are pairwise disjoint, also the sets G ′ a , a = 1, . . . , 2ℓ, have this property.
Remark 2 Because the directional distribution θ is assumed to be an even measure, in the construction above one can choose u ℓ+a = −u a , a = 1, . . . , ℓ. Then the facets f W a and f W ℓ+a are parallel.

Lemma 4
The compact convex polytope W constructed in Lemma 3 also satisfies the following property: ∀ ε > 0, ∃ t * (ε) > 0 such that the following encapsulation time relation holds, Proof 4 Let us use the notation introduced in the Lemma 3 and in its proof. For
Lemma 5 For all W ′ , all t > 0 and all ε > 0 and t * (ε) such that (23) holds, there exists t 1 = t 1 (ε, t) ∈ (0, min{t, t * (ε)}) such that for all t 2 ∈ (0, t 1 ], Notice that ζ t is monotonically increasing in t, because if at some time a cell C ′ is divided into the cells C ′′ , C ′′′ we have Then, by subadditivity of Λ, Since the process Y ∧ W ′ has no explosion, for any fixed In the sequel for t > 0 and ε > 0 the quantity t 1 = t 1 (ε, t) is the one given by Lemma 5. Recall that under the identification E with E ∧ W we can write P(Y ∈ E) = P(Y ∧ W ′ ∈ E) for all E ∈ B(T W ′ ), see (5). This identification also allows us to put for all E ∈ B(T W ′ ), D ∈ B(T W c ), s > 0: Lemma 6 For all t > 0, ε > 0 and t 2 ∈ (0, t 1 ], we have Then, By using the relation |P(Γ) − P(Θ)| ≤ P(Γ∆Θ), the result follows.
In the following results W is a window such that W ′ ⊂ int(W ). We use the notation S = S(W ′ , W ) for the encapsulation time.
Proposition 1 For all t > 0, ε > 0 and t 2 ∈ (0, t 1 ], we have Proof 7 Let us first show, Let Y ′ = (Y ′ m : m ∈ N) be a sequence of independent copies of Y , and also independent of Y . Relation (3) yields On the event S = s we have W ′ ⊆ C 1 s , this last is the cell containing the origin at time s, and thus where in the last inequality we use Lemma 6. Corollary 1 For all t > 0, ε > 0, t 2 ∈ (0, t 1 ] and all s ∈ (0, t 2 ) it holds Lemma 7 For all t > 0, ε > 0 and t 2 ∈ (0, t 1 ] we have for all D ∈ B(T W ′ ), E ∈ B(T W c ), Proof 8 Firstly, we show the following conditional independence property, We use the notation introduced in the proof of Proposition 1 and we also shortly write E instead of E ∧ W c . Also the arguments are close to the ones in the proof of relation (25).
Recall that on the event S = s we have W ′ ⊆ C 1 s . By Y s ⊞ (Y ′m t−s : m ≥ 2) we mean that the tessellations Y ′m t−s are nested only into the cells C m s of Y s with m ≥ 2, and not into the cell C 1 s . From (25), (26) and the independence of the random variables Y s , Y ′m t−s , m ≥ 1 we obtain, Then (27) is verified. By using this equality and Corollary 1 we find, In an analogous way it is proven, which finishes the proof.
Proposition 2 yields the proof of Theorem 1, by substituting 4 ε by ε.

Comparison of STIT and Poisson hyperplane tessellations (PHT)
Intuitively, we expect a gradual difference in the mixing properties of STIT and PHT. Hyperplanes are unbounded, while the maximum faces (also referred to as I-faces) in a STIT tessellation are always bounded. But these I-faces tend to be very large (it was already shown in [10] for the planar case that the length of the typical I-segment has a finite expectation but an infinite second moment). But since we have shown that the tail σ-algebra for STIT is trivial, then STIT has a short range dependence in the sense of [2].
One aspect is the following. For PHT, Schneider and Weil [14] (Section 10.5) showed, that it is mixing if the directional distribution θ (see (22)) has zero mass on all great subspheres of S ℓ−1 + . And they contribute an example of a tessellation where this last condition is not fulfilled and which is not mixing. In contrast to this, Lachièze-Rey proved in [8] that STIT is mixing, for all θ which are not concentrated on a great subsphere.
Concerning the tail σ-algebra, we have Theorem 2 for the STIT tessellations. In contrast, for the PHT the tail σ-algebra is not trivial.
Lemma 8 Let Y PHT denote a Poisson hyperplane tessellation with intensity measure Λ which is a non-zero, locally finite and translation invariant measure on H, and it is assumed that the support set of Λ is such that there is no line in R ℓ with the property that all hyperplanes of the support are parallel to it. Then the tail σ-algebra is not trivial with respect to the distribution of Y PHT .
Proof 10 Let (W n : n ∈ N) be an increasing sequence of windows such that for all n ∈ N, W n ⊂ int W n+1 , and R ℓ = n∈N W n . Let B 1 be the unit ball in R ℓ centered at 0. Consider the following event in B(T): D := {∃ an hyperplane in Y PHT which intersects B 1 }.
Because outside of any bounded window W n all the hyperplanes belonging to Y PHT can be identified and thus it can be decided (in a measurable way) whether there is a hyperplane which intersects B 1 , we have that D ∈ B(T W c n ) for all n ∈ N, and hence D ∈ B −∞ (T). We have P(D) = 1 − e −Λ([B1]) , and this probability is neither 0 nor 1.