Crack STIT Tessellations: Characterization of Stationary Random Tessellations Stable with
Respect to Iteration
Author(s): Werner Nagel and Viola Weiss
Reviewed work(s):
Source: Advances in Applied Probability, Vol. 37, No. 4 (Dec., 2005), pp. 859-883
Published by: Applied Probability Trust
Stable URL: http://www.jstor.org/stable/30037364 .
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Adv.Appl. Prob. (SGSA)37, 859-883 (2005)
Printed in NorthernIreland
© AppliedProbabilityTrust2005
CRACK STIT TESSELLATIONS:
CHARACTERIZATION
OF STATIONARY RANDOM TESSELLATIONS STABLE
WITH RESPECT TO ITERATION
Jena
WERNERNAGEL,*Friedrich-Schiller-Universitdt
VIOLAWEISS,**FachhochschuleJena
Abstract
Our main result is the proof of the existence of random stationarytessellations in ddimensional Euclidean space with the following stability property: their distribution
is invariantwith respect to the operationof iteration(or nesting) of tessellations with
an appropriaterescaling. This operation means that the cells of a given tessellation
are individually and independentlysubdividedby independent,identically distributed
tessellations, resulting in a new tessellation. It is also shown that, for any stationary
tessellation, the sequence that is generated by repeated rescaled iteration converges
weakly to such a stable tessellation;thus, the class of all stable stationarytessellationsis
fully characterized.
Keywords:Stochastic geometry; randomtessellation; iterationof tessellations; nesting
of tessellations;weak convergence;stabilityof distributions
2000 MathematicsSubjectClassification:Primary60D05
Secondary60G55
1. Introduction
In this paper, we introduce a new model for random stationary tessellations (where by
'stationary' we mean that their distributions are translation invariant) in the d-dimensional
Euclidean space Rd, d> 1. A mathematical motivation for these tessellations comes from the
consideration of iteration (or nesting) of tessellations. The operation of iteration generates a
new tessellation I(Yo, y) from a 'frame' tessellation Yo and a sequence / = {Y1, Y2, ... } of
independent, identically distributed (i.i.d.) tessellations by subdividing the ith cell, pi, of Yo
by intersecting it with the cells of Yi, i = 1, 2,.... This operation of iteration can be applied
repeatedly, and combined with an appropriate rescaling. The problem arises as to whether there
exists a limit tessellation when the number of repetitions goes to infinity. A further question is
how such limit tessellations can be described if they exist. This problem was posed to one of
the authors by R. V. Ambartzumian in the 1980s. A related question was stated in [3].
A key notion in the investigation of limits is that of stability with respect to an operation.
In a previous paper [10], the authors showed that a stationary tessellation can appear as a limit
(in the sense of weak convergence) of repeated rescaled iteration if and only if it is stable with
respect to iteration (STIT). We also listed some properties of STIT tessellations, but the main
problem of their existence remained open. (The 'tentative construction' described in Section 5
of [10] fails to demonstrate this.)
Received 19 January2005; revision received 8 June 2005.
* Postaladdress:Friedrich-Schiller-Universitit
Jena,FakultitfiirMathematikundInformatik,D-07737 Jena,Germany.
Email address:[email protected]
** Postal address:FachhochschuleJena, D-07703 Jena, Germany.
859
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860 * SGSA
W. NAGELAND V. WEISS
In this paper, we describe a construction for all stationary random tessellations that are
STIT. Furthermore, it is shown that all sequences of rescaled repeated iterations of stationary
tessellations converge to a tessellation of the above-mentioned class. Thus, the domains of
attraction of the STIT tessellations are also characterized.
Concerning potential applications, these STIT tessellations can probably be good approximations for real tessellations with 'hierarchical' structures, which can be observed in some
crack or fracture structures, e.g. the so-called craquelde on pottery surfaces (see [11]).
The definitions of iteration and of STIT tessellations are recalled in Section 3. In Section 2,
the model, referred to as a crack STIT tessellation, is introduced and its construction inside
a bounded window W is described in detail. A key property of this construction is given in
Lemma 2. This property looks rather technical, but provides the basis for the proof of the main
result, namely Theorem 2. Section 4 is devoted to the study of the capacity functional of the
random closed set of all boundary points of the cells of the crack STIT tessellation. These results
are of their own interest, but in the context of the present paper they serve as a useful basis for
several proofs. In particular, the limit of a sequence of capacity functionals corresponds to the
limit in the sense of weak convergence of random closed sets. Finally, in Section 6, the domains
of attraction of crack STIT tessellations are described as the sets of all stationary tessellations
in IRdthat have the same parameters Sv and R. For a stationary tessellation, the mean total
(d - 1)-volume of the set of boundary points of cells per unit d-volume is Sv, and ,R is the
directional distribution ('rose of directions') of the cell boundaries at a typical point (see [15]
for exact definitions). The principal tool in the proof is a generalized version of Korolyuk's
theorem, which is well known for point processes on the real axis. It is a consequence of this
result that the crack STIT tessellations described in Section 2 are the only stationary STIT
tessellations that exist.
2. Description of the model for crack tessellations in a bounded window
Let IWddenote the d-dimensional Euclidean space, N the set of natural numbers, and [3J, FS]
the measurable space of all hyperplanes in IRd, where the a-algebra is induced by the Borel
a-algebra on a parameter space for JM.For a set A C IRd,we write
[A] = {ge ~C: g nA
0)}.
For a hyperplane g e J, denote by g+l and g-1 the two closed half-spaces that are generated
by g. Let A be a locally finite and translation-invariant measure on [M, 3)].In order to guarantee
that tessellations with bounded cells are generated by the construction which is described later,
it is assumed that A is not concentrated in too few directions, i.e. we will assume that the
following condition holds in all sections of the paper.
E 3J with linearly independent normal
Condition 1. There exist hyperplanes gi,...,gd
directions and {g, ... , gdj} supp(A), where supp(A) denotes the support of A.
In this section, W C IR is a fixed compact set, sometimes referred to as the window. (For the
sake of simplicity the reader may choose, e.g. W = [-1, l]d, 1 > 0.) An essential assumption
is that
0 < A([W]) < o .
(1)
We introduce the construction of the tessellations in such a bounded window; in Section 5, it
will be shown that their distribution does not depend on the choice of W.
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SGSA
Crack STIT tessellations
861
2.1. Intuitive ideas and description using an algorithm
The intuitiveidea of the constructionis as follows. The window W has an exponentially
distributed'lifetime'. At the end of this time intervala randomhyperplaneyl is introduced
into W, anddivides W into two new 'cells'. These two cells haveindependentandexponentially
distributedlifetimes until they are divided by furtherrandomhyperplanes.After any division,
the exponentiallydistributedlifetimes of the new cells begin, and they are independentof all
the otherlifetimes. Special attentionmust be paid to the adjustmentof the parametersof these
exponentialdistributions. Roughly speaking, these parametersare related to the size of the
respectivecells such thatsmallercells have stochasticallylongerlives thando largerones. This
procedureof repeatedcell division is stoppedat a fixed time a > 0 and the state at this time is
interpretedas a realizationof the tessellation R(a, W) as a set of randompolytopes.
Fora betterunderstandingandpictureof the constructionof this tessellation,we startwith a
descriptionof R(a, W) as the outputof an algorithm.In this algorithm,we use an auxiliaryset,
T, of pairs (r, W'), where t is the time thathas alreadyexpiredand W' C W is a set that has
still to be treated,i.e. which is probablysubdividedfurther.An i.i.d. sequence {(rj, yj)}, j =
1, 2, ..., of pairsof independentrandomvariablesis given. Here, rj is exponentiallydistributed
with parameterA([W]) and yj is a randomhyperplanewith distributionA([W])-1 A(. n [W]).
Algorithm (a, W, A). Initialize as follows: j = 0, T = {(0, W)}, R = 0.
until T = 0 for (, W') E T do
+ 1
(i) j j
(ii) if T +
Tj
< a then
(a) if yj e [W'] then T = (T\{(T,
W')})U{(r + j, W'nyI),
(r+ry,
W'
y 1)}
(b) else T = (T \ {(r, W')}) U {(r + rj, W')}
(iii) else T = T \ {(r, W')}, R = R U {W'}
end
The outputof the algorithmis R (a, W) = R, which is a set of randomconvex polytopes in
W if W itself is a convex polytope. An exampleof a realizationfor d = 2 anda discretemeasure
A thatgives equal weights to the horizontaland verticaldirectionsis shown in Figure 1.
Foranytwo pairs(r', W'), (r", W") e T, we have (int W')n (int W") = 0, i.e. the interiors
of W' and W" are disjoint. Hence, the distributionof the outputdoes not dependon the order
in which the elements of T are treatedin the for-loop of the algorithm.
The algorithmcorrespondsto the following intuitive interpretationof the developmentof
R(a, W). Considerthe rj to be the lengths of time intervals. The sets W' featuringin T have
exponentiallydistributedlifetimes until they are divided by a hyperplaneyj. At the moment
of division, the independentlifetimes of the two new cells W' n y+1 and W' ny
begin.
Althoughthe randomvariablesrj are identicallydistributed,the rejectionprinciple,expressed
by the condition 'if yj E [W']', yields a lifetime for W' that,except for those W' which appear
in the resultingoutputR(a, W), i.e. when the time bound a is reachedduringthe lifetime of a
cell, is exponentiallydistributedwith parameterA([W']).
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W. NAGELAND V. WEISS
862 * SGSA
in algorithm
FIGURE
1: An example(providedby JoachimOhser)of a realizationof the construction
(a, W,A), whereA is concentrated
andverticaldirections.
equallyin thehorizontal
2.2. Formal description
We start the formal descriptionwith a sequence {(rj, yj)} of i.i.d. pairs of independent
randomvariables rj and yj, j = 1, 2,..., where the tj are exponentiallydistributedwith
parameterA([W]) and the yj arerandomhyperplaneswith distributionA([W])-1A(. n [W]).
For any a > 0, we will define a deterministicfunction that maps the sequence {(j, yj)} to a
set of convex polytopes which intersect W.
Forthe sake of convenience,we will use the terminologyof graphtheory. Forthe definitions
of the following notions, see, e.g. [7] or [6]. Denote by B the following infinite-levelbinary
tree, in which all nodes have two children. The set of nodes is {(rj, yj, s(j)), j = 1, 2, ... },
with s(j) e {-1, +1} and the conventionthat the root of B is (ri, yi, 1). On the next level,
the 'left-hand'child is (T2,y2, -1) and the 'right-hand'child is (r3, 73, 1). In this canonical
way the nodes are denotedlevel by level and, on each level, from left to right. The nonrandom
parameters(j) has the value -1 for all left-handchildrenand +1 for all right-handchildren.
For a real a > 0, the randomtree B(a) is defined as the (almost surely) finite subtreeof B,
with the same root (rt, yi, 1), that has (zrjk+l, Yk1, s (jk+1)) as a leaf if and only if
k
k--+1
^ji <a
i=1
ji(2)
i=1
for the path (rT, yi, 1), (r12, yj2,s(j2)),...,
(Tjk+, yjk+, S(jk+1)). Here and in the following,
jl = 1 and s(jl) = s(l) = 1. If rt > a then B(a) is the tree with the root (ri, yi, 1) as its
only node. We will typicallywrite a pathin B(a) from the root to a leaf in the abbreviatedform
(s(jl), ..., s(jk+l)), i.e. only indicatingthe indices along this path and whetherthe steps are
going to the left-handor to the right-handchild. This is sufficientfor the determinationof a
path in a binarytree.
As in the algorithmabove, we will interpretrtj as the time intervalbetweenthe introductions
of the randomhyperplanesy, _ and yji into the window W. The intervaltr is the time elapsed
before the first hyperplaneyl is introduced. Thus, the tree B(a) contains all paths which
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CrackSTITtessellations
SGSA
863
(1)
(4)
(6)
(5)
(12)
(7)
(14)
(13)
(28)
(15)
(29)
(58)
(59)
FIGURE
2: An exampleof a subtreeB(a) of B. Thelengthsof thepathsfromtherootto theleavesare
determined
to thestructure
givenin Figure3 andalsoto the
by condition(2). SubtreeB(a) corresponds
schemein Figure4 (seebelow).
characterizethe developmentup to time a. An illustrativeexampleis given in Figure2. Denote
the set of all pathsin B(a) from the root to the leaves by
/f(a) = {(s(jl), ...,
s(jk+))
{+1
k >_0, (s(jl),...,
x {-1, +1}k:
s(jk+l)) is a pathto a leaf in B(a)}.
We will use the abbreviatednotationys(ji+l) for yi(j) . Notice thatthese half-spacesare
specified by yji and the parameters(ji+l) associated with the next node in the path. For an
a > 0, we define the set R(a, W) = {W} if f/(a) = {(1)}, i.e. if Tr > a; otherwise,
R(a, W) =
ys(ji+)
(
n W: (s(ji) ...,
Note that there can occur paths (s(jl),...,
Yjl+l
s(jk+)) e f(a)
\ {0}.
(3)
s(jk+1)) e f(a) with
iys(ji+i)nW],
1,5
k -1,
and, thus, yji+, does not divide the cell. Hence,
ys(ji+1)
nW
n ys(jl+2)
i=l
is either empty or does not differ from the undividedpreviouscell ni=l ys (ji+) n W. This
means thatsuch a Yjl+Ican be rejectedor ignoredin the construction;for the formalismin our
proofs, it is more convenientto include it in the description.
Obviously,R(a, W) is a randomset of convex polytopes, intersectedwith W. We will typically understanda randomtessellation as the randomclosed set of the union of the boundaries
of its polyhedralcells, exclusive of the boundaryof W. Accordingly,for an a > 0, we introduce
the set
Y(a,W)=cl((
U
aw') \w)
W'eR(a,W)
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(4)
W. NAGELAND V. WEISS
864 SGSA
Y2
Y7
Y29
Y14
Y3
Y1
Y6
FIGURE
3: A realizationof Y(a, W) that correspondsto the tree B(a), shown in Figure 2. Note that not
necessarily all nodes of B(a) yield an edge: e.g. if yl4 fell to the left of Y1 then it would have to be
ignored. Consequently,in this case y29 would extend from yl to the right-handboundaryof the window
W (dashedline).
where aW' denotes the boundaryof the set W' and 'cl' the topological closure of a set. An
example is shown in Figure 3.
Later in the paper,we will provide relationsbetween the tessellations for differentvalues
of a. By convention,we choose a = 1.
Definition 1. The tessellationY(1, W) is called the crackSTITtessellationfor the measureA
in W.
Although in this definitionthe term 'STIT' is used, an exact definitionof stability will be
postponed to Section 3, and it will be the main task of the following sections to prove that
Y(1, W) is indeed a tessellation stable with respect to iteration. Furthermore,we will show
thatits distributionis consistent,i.e. that Y(1, W) can be understoodas the restrictionto W of
a stationaryrandomtessellationof IRd.The set R (1, W) is the set of cells of Y(1, W).
Algorithm(a, W, A) can be relatedto the formaldescriptionwith the help of binarytrees, in
the following way. Follow all paths (s(j), ... , s(jk+l)) e 13(a)and performthe intersection
stepwise as
ys(ji+)
y
n
W
i=1
for 1 = 1, 2, ..., k. If such an intersection is empty for some 1, then this path is not followed
furthersince it will not contributeto the set of cells in R(a, W). At the end of those paths that
do not yield an empty intersection,the resultingpolytopes occur. Since both the rj and the yj
are i.i.d., the orderof indices which are used in the algorithmmust not be the same as in the
tree B.
The union of boundariesY(1, W), or the set of polytopes R(1, W), indeed representsa
tessellation in W since, accordingto (3), the elements of R(1, W) have disjoint interiorsand
fill W, since in the tree B(l) a node eitherhas both right-handand left-handchildren(with the
s-values + 1 and - 1, respectively)oris a leaf withoutchildren.Thispropertycan also be derived
from the algorithm.We now show thatthe tessellationis almost surely (a.s.) nondegenerate.
Lemma 1. If 0 < A([W]) < 00 then,for all a > 0, R(a, W) is a.s. finite.
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CrackSTITtessellations
SGSA
Proof The process Xt = card(8(t)),
865
t > 0, of the number of paths in f (t) is a Furry-Yule
or linearbirthprocess (see [5] or [13]) with parameterA([W]), i.e. with birthrates A([W])Xt.
As is well known (see [5]), the randomvariableXt has a geometricdistributionwith parameter
exp{-A([W])t)}, which implies that
P(Xt < oo) = 1
for all t > 0.
The definitiongiven in (3) yields card(R(a, W)) < card(f(a)) and, thus,
P(card(R(a, W)) < oo) > P(card(3(a)) < oo) = 1.
Thereis an obvious relationshowinghow the length,a, of the time intervalandthe intensity
commute. If Tj is exponentiallydistributedwith parameterA ([W]), thenc-1 tj is exponentially
distributedwith parametercA([W]), for any c > 0. Since the above-defineddistributionof
the hyperplanesyj is invariantwith respectto constantprefactorsc for A, we can immediately
derivethat,for a given A, RA(a, W) := R(a, W) has the same distributionas RcA(c-la, W),
i.e. the constructionwith measurecA and time intervalof length c-1a.
3. The crucial relation between the iteration of tessellations and the construction
For tessellations, the operationof iteration(also referredto in the literatureas nesting) is
definedas follows. Let Y1,Y2,... be a sequenceof i.i.d. stationarytessellationsin IRdandwrite
y = { Yi, Y2, ... }. Furthermore, assume that Yo is a stationary tessellation that is independent
of #. For this definition,it is useful to considerthe set of cells (which are convex polytopes),
C(Y), of a tessellationY. Assume thatthese cells arenumberedandthatC(Yo) = {pl, p2 ... }.
The iterationof the tessellation Yoand the sequence ' is defined as the tessellation
I(Yo, ) = YoU
U
pi
(5)
(Yipi).
C(Yo)
This formula describes the operationin terms of the boundariesof the cells. For the cells
themselves, it means that the cells pi of the so-called frametessellation Yoare independently
subdividedby the cells pik, k = 1, 2,.... (or faces) of the tessellations Yi that intersectthe
interiorof pi. A list of referencesconcerningiterationcan be found in [10].
For a real numberr > 0, the tessellation r Y is generatedby transformingall points x e Y
into rx. Accordingly,r y denotes the applicationof this transformationto all tessellations of
the sequence /. Let Sv be the mean total (d - 1)-volumeof the set of boundarypoints of cells
per unit d-volume (surfaceintensity)of any of the tessellations Yo,Yi,.... Then I (Yo, ') has
the surfaceintensity2Sv and I (2Yo,2') has the surfaceintensity Sv.
Let Yo be a stationary tessellation and
1, 2, .... a sequence of sequences of tessellations
suchthatthe tessellationsinvolved(includingYo)arei.i.d. Thenthe sequence12(Y0),13(Y0), *..
of rescalediterationsis definedby (see [10])
I2(Y0) = I (2Yo,2i1),
Im(Yo)= I(mYo, mi, ...,
= I(I(mYo, m 1,...,
m
Im-1)
m'm-2),
mjm-1),
m = 3, 4,....
Here, m is the rescaling factor, which is chosen to keep the parameterSv of the tessellation
Im(Yo)constant for all m. Later, in Section 6, it will be shown that this also yields the
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W. NAGELAND V. WEISS
866 * SGSA
convergenceof Im(Yo), as m -+ oo, to a nondegeneratetessellation. We use the abbreviation
Im(Yo) since it is assumed that all the other tessellations in the sequences Y1, Y2.... are
independentand have the same distributionas Yo.
The symbol '-' is used to denote equalityof distributions.
Definition 2. A stationarytessellation Y is said to be stable with respectto iteration(STIT) if
Y = Im(Y) for all m= 2, 3,...,
i.e. if its distributionis not changedby repeatedrescalediterationwith sequencesof tessellations
with the same distribution.
This stability concept is related to the stability of the union of randomclosed sets. The
operationof iteration,as definedin (5), is essentially a union of partsof (d - 1)-dimensional
mosaic faces. Hence, it can be seen, in close analogy to the results of [8, Proposition3-5-1,
p. 67], thatfor STITtessellationsthe scaling exponentis necessarily 1, i.e. m1 is essentially the
only scaling factorthatguaranteesconvergence.
The cruxof this section is (7), which providesa key propertyof the tessellationsdescribedin
Section 2: iterationof these tessellationscommuteswith the additionof time intervalsfor the
construction. This feature will allow us to give an elegant proof that the crack tessellations
are indeed STIT. The proof of Lemma 2 is essentially based on the 'lack of memory' of
the exponentialdistributions,which implies that (R(t, W))t>o and (Y(t, W))t>o are Markov
processes.
Here we denote the elements of R(a, W) by Wu, u e N.
Lemma 2. Let R(a, W), R1(b, W), R2(b, W), ... be independentrandomsets of polytopes
(intersectedwith W) definedaccording to (3), with a, b > 0. Then
U
{W n Wui: Wi
Wu int Wui Z
E Ru(b, W), int
R(a + b, W).
0)}
(6)
WuGR(a,W)
For independenttessellations Yo(a, W), Yi(b, W), Y2(b,W),..., as definedin (4), and
y(b, W) = {Yi(b, W), Y2(b, W), ...},
we have
(7)
I(Yo(a, W), y(b, W)) o Y(a + b, W).
Proof Due to the independence of the rj and yj, j = 1, 2 .. .,and the lack of memory of the
exponential distribution, for all a > 0, t > 0, 1 = 0, 1,..., and all paths (s(jl), ... , s(jl+l))
in B we have
(1+1
1
i=1
i=1
ji
Tj)1+1=p
a<
>
i=1
Write fP (b) for i.i.d. copies of f (b) indexedby p e f(a). For a given path (s(jl), s(j2),...
in B, denote by 1(a) the randomindex for which
1(a)
l(a)+l
i=1
i=1
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)
SGSA* 867
Crack STIT tessellations
"15
Wn
y+l
n y3l
59
T29
Y7
z 14
Wn y+1
3
Y2
14
58
14
T28
V3
V13
6
Y6
Wn yl n y31
W
r12
Y1
1
WN yll ny2+1
t5
Y2
72
2
Wn yl 1
T4
Wnylnl
y2-1
a
0
a+b
FIGURE
4: Theschemeof timeintervalsupto timea. Thiscorresponds
to theexampleof thetreeB(a)
segmentsareproportional
in Figure2. Herethelengthsof thehorizontal
to ther-values.
Note that the cases k = 0 and Ti > a, i.e. 1(a) = 0, are also included in the following
considerations.We then have
R(a + b, W)
=
s((a))...,
(i yS(i+i)) n w: (s(j)....
i=1
1(a)
=
U
(s(ji), ..., s(jl(a)+l))Ef
fn
(a)
s(jk+1)) e f(a
k
sj(iil)n
i=l
ys(j+)n w:
n
i=l(a)+l
(s(ji), ...,
s(jl(a)),....,
U
k'
{(oys(ji+li)n
(a)
i=1
ys(;+):
w) n
i=1
(s(j'),...,s(j,+l))e
U
P/(a + b)} \ {0}
s(jk1))
1(a)
p=(s(jl),...,s(jl(a)+l))€
b)I \ {0}
{(Wun Wui: Wuie Ru(b, W), int Wun int Wui
3
) \{0}
0}.
WuR(a,W)
Thus, (6) holds.
An interpretationof the above derivationis as follows (see Figure 4 for an illustration).
Observethe process of constructionat time a. On each path (of the tree B(a + b)) the time
interval from a to the introduction of the next hyperplane is E(a
Ty, - a. This has the
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868 * SGSA
W. NAGELAND V. WEISS
same distribution(exponentialwith parameterA([W])) as r.: for any index j'. Hence, the
distributionof the result is the same for both methods: either continue the constructionat
time a with the 'remainder'of the lifetime of each cell, or at time a starta new lifetime tij and
thus startthe constructionof a set Ru(b, W) n Wu,where Wuis a polytope in R(a, W). In the
terminologyof trees, this lattermethodmeansthat,at each leaf u of B(a), an independentcopy
Bu(b) of B(b) is appendedby replacingthis leaf by the root of Bu(b) but keeping the s-value
of u.
In order to prove (7), consider the boundariesof the polytopes in correspondingsets R.
Denote by Yo(a, W) and
b, W) = (Yl(b, W), Y2(b, W),...)
independent copies of tessellations with the same distributionsas Y(a, W) and Y(b, W),
respectively.Thus, using (5), we have
I(Yo(a, W), y(b, W)) = Yo(a, W)U
(Yu(b, W) n Wu) D Y(a +b, W).
U
WuR(a, W)
In a preliminarymanuscriptfor the presentpaper,the authorsconstructeda very detailed
proof of Lemma 2 by consideringthe joint distributionsof paths of the trees describedabove.
Thereare no difficultiesin it and it follows straightforwardlyalong the lines of the proof given
here.
4. Properties of the capacity functional of Y(a, W)
Let C denote the set of all compact subsets of Rd. The distributionof the randomclosed
set Y(a, W) is uniquely determinedby its capacity functional TY(a,W): - + [0, 1], which is
defined as
Ty(a,W)(C) = P(Y(a, W) n C # 0)
for C e C.
We will not give a generalexplicit formulafor this capacityfunctionalhere, but will express
some of its featuresthat are useful in the proofs in Sections 5 and 6.
Lemma 3. IfC e C is connected,with C C W, then,for a > 0,
1 - TY(a,W)(C) = P(Y(a, W) 0 C = 0) = e-aA([]).
Proof For k > 1, with the i.i.d. assumptionfor the yj and the abbreviation
(s(ji),...s(jk+l))
l){1}
x {-1,+1}k
firstnote thatwe obtain
(k)
k- 1
(k-l)
k
P(C C
ys(ji+i))
P( CC
=
ys(ji1))P(Yjk
[C])
i=1
i=1
= P(yl i [C])k
S(A([W])- A([C]) k
=
A([W])
I
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CrackSTITtessellations
869
by recursion. Recall the following well-known relationbetween the Poisson and exponential
distributions.For the given parametersit is writtenas
k+l
a
= (aA([W]))ke-aA([W]),
k = 0, ,..
i=1
i=1
Thus, owing to the independenceassumptions,for a connected set C C W we have
P(Y(a, W)n C = 0)
{W}, Y(a, W) n C = 0)
= P(R(a, W) = {W}) + P(R(a, W)
< a and thereexists a k > 1 and a path
= P(ri > a) + P(
\k
(s(i).
\
. . s(jk+)) e
(a) such that C C
ys(ji+1))
i=1
k
o00 (k)
= P(
> a)+±
E
S-aA([W])
k
k=l
i P CC
a <
r
P
k=l
ys(j i+i)
i=1
i=1
i=
-aA([WI) A([W])-A([C])
A([W])
(aA([W]))k
k!
/
k
k+1
k
= e-aA([C]).
Let Co denote the set of all compact sets with finitely many connected components. For
C e Co, which consists of more than one connected component, we now derive a recursion
2, where the Ci e C are connected and
formula. For C e Co with C = Uki=lC, k
Ci n Cj = 0 for i : j, define the sets
{Zi, Z2=
UCi
ieJ1
U
i
(8)
Ci
{1,...,k}\Ji
for all nonemptysets J1 C {1, ..., k} with Ji
{1,.... k}. In the notationfor these sets we
omit referenceto Ji, and symbolically write z1,Z2for the sum over all such sets. (Notice
that these are not orderedpairs!) Denote by [Z1I Z2] the set of all hyperplanesthat separate
Z1 and Z2. By conv C we denote the convex hull of C.
Lemma 4. If C e Co has more than one connectedcomponentand C C W, then,for a > 0,
P(Y(a, W) n C = 0) = e-aA([con C]) +
aA([Zi
I Z2])
dte-taA([con C])
ZI,Z2
x P(Y(a(1 - t), W) n Zi = 0) P(Y(a(1 - t), W) n Z2 = 0).
Proof The sets Z1 and Z2, which are defined as in (8), are separatedfor the first time by
the hyperplane yji if there is a path (s(jl),...,s(jl+1))
in B(a) such that C = Z1 U Z2 C
fi=1 ys(ji+i), with
l-1
Zi C yf ys(ji+)
i=1
l-1
n ys(j+l),
Z2 C n
ys(i+1)
Y(-S(jl+1)),
i=1
or if the same expressionshold with Zi and Z2 interchanged.
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W. NAGELAND V. WEISS
870 * SGSA
If we assume thatsuch a pathexists, we can focus our attentionon the subtreesof B(a) with
the roots s(jl+l)
The subtree with the root s(jl+l) generates - according to (3)
and -s(jl+1).
and (4) - a tessellationinside n1=1 ys(ji+l) thathas the same distributionas
Y(a -
rji , W) n nys(ji+l).
i=1
i=1
With these preliminaryconsiderations,we are able to prove the lemma. To do this, we
partitionthe event Y(a, W) n C = 0 with respectto all partitions{Z1, Z2} of C into two parts,
and with respect to the length I of the path in B(a) when Z1 and Z2 are separatedfor the first
time. When we use indices such as those in Yi(', W) and Y2(-,W), we assume that these two
tessellationsare independent.Withthe substitutionsr =
i= ti and t = r/a, the formulafor
the volume of an (1 - 1)-dimensionalsimplex, and with the abbreviationL-() as in the proof
of Lemma 3, this yields
P(Y(a, W) n C = 0)
= P(Y(a, W) n conv C = 0) + P(Y(a, W) n conv C # 0, Y(a, W) N C = 0)
-
e-A
([convC])
(1)
oo
a
E
+-E
a
dtl - --
1l
/
{
dt (A([W]))le-(A([W])
\
ti
l ti) l[oai
i=1
Z1,Z2 /=1
/-1
x P(conv C
ys(ji+i))
P(Yj e [ZiI Z2])
i=1
x P(Y(a -
ti, W) n Z2= )
tji,w) nZi = ) P(Y(a i=1
i=1
- e-aA([conv C])
+
3
L
A([Z I Z2])
dr e-rA([W])
Z1 ,Z2
x P(Y(a - r, W) n Z1 = 0) P(Y(a - r, W) n Z2 = 0)
0r
0
x
(A([W]) - A([conv C]))/-1
for
dt
.
-1
dt-1 l[O,r]
(
= e-aA([conv C])
+
C
aA([Z1I Z2])
dt{e-taA([conv
])
Z1,Z2
x P(Y(a(1 - t), W) n Z1 = 0) P(Y(a(1 - t), W) n Z2 = 0)}.
Here, 1[.] is an indicatorfunction.
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\
ti)
CrackSTITtessellations
SGSA
871
Lemmas 3 and 4 and a straightforwardcalculationyield the following corollary.
Corollary 1. If C = Ci U C2 e (0 has exactly two disjoint connected components,C1 and
C2, andC C W, then,for a > 0,
P(Y(a, W) n C = 0)
e-aA([conv C]) +
x
=-
A([C1 I C2])
e-aA([conv C]) _ e-aA([C1])e-aA([C2])
A([C1]) + A([C2]) - A([conv C])
e-aA([convC]) +
if the denominatoris nonzero,
aA([C1 I C2])e-aA([C1])e-aA([C2])
otherwise.
Thus, the values of the capacity functional Ty(a,w) can be calculatedby recursionfor all
C
Co0.
5. Consistency, stationarity, and STIT
The calculationsin the previous section show that, for any window W and C e Co with
C C W, the value of the capacity functional TY(a,w)(C) is independentof W. This implies a
consistencypropertyand,hence, the existence of a correspondingtessellationY(a) of all of R~d
Theorem 1. If the measure A on [sf, Si] is invariantwith respect to translationsof Rd and
Condition1 holds, then,for any a > 0, thereexists a randomtessellation Y(a) ofWRd
such that
Y(a) n W = Y(a, W) for all compactwindows W satisfying (1).
Proof Let a > 0 and WI, I = 1, 2,....
ie N and I > i, we have
be a sequence of windows W1= [-1, l]d. For any
Y(a, Wi) n Wi = Y(a, Wi)
since, due to Lemmas 3 and 4, for all C e Gowith C C Wi,
T(Y(a,w)nwi)(C)
= TY(a,WI)(C) = TY(a,Wi)(C).
Application of the consistency theorem Satz 2.3.1 of [15] reveals that there exists a random
closed set Y(a) in Rd such that
Y(a) n W = Y(a, Wi) for all l e N.
Now let W C IRdbe an arbitrarycompact set. There exists an 1 e N such that WI D W and,
hence,
Y(a) nW = (Y(a) n Wi) n W = Y(a, W1) n W = Y(a, W).
Lemma 5. If the measure A on [3', Si] is invariantwith respect to translations of Rd and
Condition1 holds, then,for a > 0 and the tessellation Y(a) of Theorem1,
(i) Y(a) is stationary(i.e. its distributionis translationinvariant),and
(ii) Y(a) = (1/a)Y(1).
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872 * SGSA
W. NAGELAND V. WEISS
Proof (i) For any C e Co and x e Rd, there exists a compact window W C Rd with
C U (C + x) C W. Hence, using Lemmas 3 and 4, we find that
TY(a)(C + x) = Ty(a)nw(C + x) = Ty(a)nw(C) = Ty(a)(C).
(ii) The translationinvarianceof A implies a factorizationanalogousto thatin (13), below, and
this yields A([aC]) = aA([C]) for C e Co and a > 0. There is a compact window W with
C U aC C W and, hence, using Lemmas 3 and 4, we have
Ty(a)(C) = Ty(a)nw(C) = Ty(1)nw(aC) = Ty(1)(aC) = T(1/a)Y(1)(C).
Theorem 2. If the measure A on [J, Si] is invariantwith respect to translationsof Wdand
Condition1 holds, then the tessellation Y(1) of Theorem1 is STIT
Proof Let Y(a) and y (b) be (or contain)independenttessellationsof the type involved in
Theorem 1, let a, b > 0 and C E Co, and let W D C. Then, using Lemma 2 and (7), we have
TI(Y(a),y(b))(C)
= TI(Y(a),y(b))nw(C)
= TI(Y(a)nW,y(b)w)(C)
= TI(Y(a,W),y(b,W))(C)
= TY(a+b,W)(C)
= TY(a+b)nW(C)
= TY(a+b)(C).
Now, for any m > 2 and a > 0, let Y(a), i(a),...., Ym-l(a) be i.i.d. sequences of
tessellations of the type involved in Theorem 1. Then
Im(Y(1)) = I(mY(1), mYi(1), ....
D
(L)
_OI Y
,
m
l -
mm-l(1))
, ...,
m
m-1
m
m-1
i=o
= Y(1).
It is obvious that, for all a > 0, the tessellation Y(a) is STIT.Also, in the case that a
translation-invariant
measureA does not satisfy Condition1, a STITpropertycan be statedfor
the randomclosed set Y(1) (which is generatedaccordingto (4) or Algorithm (a, W, A)) if
iterationfor hyperplaneprocesses is definedin analogy to thatfor tessellations.
6. Domains of attraction
Up to now, we have describedthe crack STIT tessellations as a class of stationarySTIT
tessellation. Using Theorem2 of [10], we conclude that these tessellationscan occur as weak
limits of repeatedrescaled iterationsof i.i.d. tessellations. We will now describe the domains
of attraction,i.e. those classes of stationarytessellationfor which the limit of repeatediteration
is such a crack STIT tessellation.
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CrackSTITtessellations
SGSA
873
6.1. A generalization of Korolyuk's theorem
Korolyuk's theorem (see [4, Chapter3, pp. 44-46]) is well known for stationarypoint
processes on the real axis. In [10], the authorsproved a version of it for stationarysegment
processes in the plane. On this foundation,it is now easy to generalizeit to processes of facets
of cells of stationarytessellationsin Rd. This result,which is given in Lemma6, is essentialin
our context, since it describesthe asymptoticeffect of rescalingthe tessellations.
Let Fd-1l denote the set of all (d - 1)-dimensional(but not lower-dimensional)convex
polytopes in Rd, and F0-1 the subset of all s E Fd-1 that have the origin o as the center of
theircircumscribedball.
In Section 3, we introducedC(Y), the set of all cells of tessellation Y. The process of all
(d - 1)-facets of its cells is defined as
Iy = {(x, s) E Id x Fd-1: s +x e {pi n pj: pi, Pj e C(Y)} n Fd-1}.
The definitionof ty requiresparticularattentionsince the cells are not necessarily 'face-toface' ('seitentreu'in the terminology of [15, p. 235]). If Y is a stationarytessellation then
Py can be consideredto be a stationarymarkedpoint process on Rd with markspace F0d-1
Stationarityimplies that its intensity measure, O say, can be factorized,i.e. that there exist a
constant Nd-1 and a probabilitymeasureK on Fd-1 (endowed with the standarda-algebra)
such that,for all measurablefunctions
[0, oo),
f: Rd x Fd-1 we have
f
(d(x,s))f(x,s) = Nd-fdx
f K(ds)f(x,s),
wheredx denotesthe elementof the d-dimensionalLebesguemeasure;see [1] or [15, Satz 3.4.1,
pp. 89-90, or Satz 4.2.2, pp. 121-122]. We assume that
0 < Nd-1 <.
This Nd- 1 is the mean number of (d - 1)-facets of Y per unit volume. The surface intensity
Sv of Y, which was mentionedin Section 3, can be definedas
Sv=
O(d(x, s)) l[o,1
Nd-
(x)Vd-l(s)
f K(ds)Vd-l(s),
and Vd-1(s) is the (d - 1)-volume of s e F0d-. Finally, the directional distribution R on £d,
the set of all one-dimensionalsubspacesin Rd, is definedby
f R(du)g(u) = [Vd-]-1
for all measurablefunctions g: £d F0d- , where
Vd-1
f
K(ds)Vd-l(s)g(u(s))
[0, oo) and subspaces u(s) e £L orthogonalto s e
=
K(ds)Vd-1s).
This yields
Sv = Nd-1Vd-1
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874 * SGSA
For a set A C Rd, we write
(A) = {(x, s) e Id x F0d-1: (s + x) n A # 0},
and by Iy ((A)) we denote the numberof facets of Y thathit A.
Lemma 6. (Modifiedversion of Korolyuk'stheorem.) Let Y be a stationaryrandomtessellation of Rd, d > 2, with surface intensitySv, 0 < Sv < oo, and directionaldistributionR.
Then,for all C e Q,
lim mP (
-((-C
m-+oo
m
(9)
2) =0
and
lim mP
m-+oo
Y n -C
-0
m
= Sv
u)\,
,R(du)I(C,
(10)
where I (C, u) denotes the lengthof the orthogonalprojectionof C onto the one-dimensional
linear subspace u e £L.
Proof We start with some preliminarygeometric considerations. We write V for the
d-dimensionalvolume andBr for the d-dimensionalball withradiusr centeredatthe origin. For
pairs(xi, si) Rd x Fd-1, we denoteby hi E Jt the hyperplanethatcontainssi +xi, i = 1,2.
Assume that
dim((sl + xl) n (s2 + x2)) < d - 2.
If hi and h2 are paralleland separatedby a distance 1 > 0, then
(hi
Br/m) n (h2 G Br/m) = 0
for m > 2r/l. For the case hi = h2, the assumption that
dim((sl + xl) n (s2 + X2)) < d - 2
and the Steinerformulafor parallelsets yield, for t > 0,
V(((si
+ Xl)
Br/m) n ((s2 + X2) G Br/m) n Bt) <
- td-2
d-27r
where Kd-2 is the volume of the (d - 2)-dimensionalunit ball. If hi
parallel,then
V((hl G Br/m) n (h2 o Br/m) N Bt)
Kd-2 ()
- h2 and they are not
m)Isin L(hi, h2)ltd2.
Now, for a given C e e, choose an r andat such thatC C Br and (si +xl) U (s2 +X2) C Bt-r.
Then
lim mV
m- o
(si + xi)
--C n (s2 + x2) e -C
m
m
< lim mV((hi $ Br/m) n (h2 G Br/m) n Bt)
m-oo
=0.
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(11)
Crack STIT tessellations
875
SGSA
Furthermore,for s e Fd- 1, a lowerboundfor V (s (1/m)C) canbe derivedwith an application
of Fubini'stheorem,which yields
V
sV --c) \ -c)
m
m
-c,
u(S))
m
> Vd-1(s)
An upperboundis given by the volume of the circumscribedbody
x n
- convC, u(s)
m
sen
-C, u(s) ,
m
where ul(.) denotes the orthogonalcomplementof u(.). Hence,
V -C
C,u(s)
+ Vd-1(s)
m
V s@-C
m
m
< V(se
(-convC,
u-(s)
H
m
C,u()
))
m
and, thus,
lim mVs
= Vd-1(S)l(C,
-C
m
m->oo
(12)
u(s))l.
Now, up to a few necessary changes in the notation,the remainderof the proof is the same as
thatgiven in [10, p. 128]. By 1{., we denotethe indicatorfunctionwith the value 1 if the event
{.} occurs, and the value 0 otherwise. For s E F0d-1,we denote by P the Palm distributionof
ty with respectto the marks. Then the refinedCampbelltheoremfor markedpoint processes
(see [16, p. 125] or [15, p. 93]) yields
lim m P(dp) f
m-m oom
= lim
mNd
(ds) l se
-1 fK(ds)
-C
1
--C
m
dx l(s
Po(dy)
>2
+x) e -C
j
(
C - x)
m
Sm-moo
>2
m
s s'+x'
1
p((x',s'))=l
=0,
since the numberof termsin the sum that are differentfrom 0 is a.s. finite and boundedfor all
m, and for any of these termswe can apply (11). With this result, we obtain
lim mP (Iy( -C
< lim m
m-o
=0
>2
PSP(p) p(ds)l se
- )C
\m
1(
-C
m
>2
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W. NAGELAND V. WEISS
876 * SGSA
and, thus, (9). It was shown in [10] that
C # 0) - E[(Fmy((C))]I= 0.
lim ml P(mY
m-oo
Finally,the Campbelltheoremand (12) yield
lim mE[4my((C))]
m-+oo
= lim m Nd-1 Vd-1 K(ds)
f
= lim mSv
m-
=SvJ,
1
K(ds)
Vd-1
dx 1 s + x e
11
- sV C
-
C
s
(du)I(C, u)l
and, thus, (10) follows.
6.2. Weak convergence of stationary tessellations to crack STIT tessellations
Now let Y be a stationaryrandomtessellation in Rd with 0 < Sv < oo and directional
distributionR. We will derive formulae for the limit of the capacity functional of repeated
rescaled iterationsIm(Y), as m -> oo, in orderto compareit with the capacity functionalof
the crack STIT tessellation Y(1). The parametersof the tessellationhave to be relatedto the
measureA, which played a role in the constructionof the cracktessellations.
We denote by
e: Rx J - J
the function that maps each pair (t, u) to the hyperplanee(t, u) orthogonalto u E £ and at
distance It from the origin. Foruniqueness,e(t, u) n u is in the upperhalf-spaceif and only if
t > 0. For a given numberSv and a directionaldistribution<1, define a measureA(SV, R, .)
on [Jf, SI] by
f
A(Sv,
, dh)g(h) = Sv
for all measurablefunctionsg: J -
dt R(du)g(e(t, u))
(13)
[0, oo). In particular,for C e C this formulayields
A(Sv, 9, [C]) = Sf
~R(du)|I1(C, u)I.
(14)
First, consider the capacityfunctionalfor a connected compact set C. Formula(10) yields
the following result.
Lemma 7. If C
C is connectedthen
lim (1 - TIm(Y)(C)) = e-A(s'y,,[c])
m-+oo
Theorem 3. If Y is a stationaryrandomtessellation in Rd with 0 < Sv < oo and directional
distribution,l, then
lim TIm(Y)(C)= Ty(1)(C) for all C e C,
m- >oo
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Crack STIT tessellations
SGSA
877
where Y(1) is the crackSTITtessellation of Theorem1 with
A = A(Sv,
?, .).
This implies the weak convergenceof the sequence Im(Y) of tessellations to Y(1).
Proof It is sufficientto considerthe capacityfunctionalsfor sets C E Co, i.e. for compact
sets with finitely many connected components(see [10, Lemma 1]). Thus, we can prove the
theoremby induction. In this proof, we always write A for A (Sv, ~, .). For C E Co, where
k
i=1
with Ci E C connectedand
Ci n Ci' = 0
i', i, i' = 1, ..., k, and apartition{ J,...,
sets, let Z = {Z1,..., Z1}with
if i
Z = Uc,
Ji} of the index set {1,...,
=l
....
k} into l nonempty
.
ieJi
(In (8) this was introducedfor the case 1 = 2.) Denote by Z(C) the set of all such partitionsZ
of C with I E {1, ..., k}.
Laterin the proof we will use the fact that, for all Z e Z(C) and all but a finite numberof
m E N,
P(mY n conv C = 0) -
P(mY n conv S = 0) # 0.
(15)
SeZ
This can be shown as follows. For the typical cell mpo of mY, we have
mpo D cony S
if and only if
[mpo]convS D conv S,
where [A]B is the morphologicalopeningof a set A with a set B (see [14]). Thus,by the refined
Campbelltheoremand the Steiner formulafor the erosion of morphologicalopen sets (again
see [14]), P(mY n conv S = 0) can be expressed as a polynomial in m (with mean mixed
volumes as coefficients). Thus, the expressionin (15) is also a polynomial in m and has only
finitely many roots.
Furthermore,in this proof we restrictour attentionto those C E Co for which
A([conv Z]) - E
A([conv S]) f 0 for all Zj E Z e Z(C) and all Z' e Z(Zj).
(16)
SeZ'
Any C e Co thatdoes not satisfy (16) can be approximatedby its parallelsets C $ Be, with
distancee > 0. By (14), the value A([conv Zj] ( Be) is linear in e and, hence, there are only
finitely many e for which (16) is not satisfied. Thus, by a continuityargumentfor the capacity
functional(which is continuousfrom above for sequencesof compactsets; see [8] or [15]), we
conclude thatthis additionalassumptioncan be made in this proof of weak convergence.
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878 * SGSA
We will show, by induction,that for all Z e Z(C), C e e0 satisfying (16), a > 0, and
almost all m e N (i.e. with at most finitelymanyexceptions),thereexist numbersdm(Z), d(Z),
qm(C), and c(C) such that
(i)
P(Im(Y) nC=0)=
P(m n cony Zj =0)+m(C),
dm(Z)
ZeZ(C)
d( -Z
= dm(Z) forleN,
0 < qm(C) < c(C)mP(cmy((convC))
c(rC) < c(C)
(ii)
(18)
> 2),
(19)
for 0 < r < 1,
(20)
d(Z)
(21)
P(Y(a) n C = 0) =
e-aA([cnZj])
ZjeZ
ZEZ(C)
(iii)
(17)
ZjeZ
lim dm(Z) = d(Z),
(22)
lim qm(C) = 0.
(23)
m->oo
m-+oo
Proof of (i) by inductionover the numberk of connected componentsof C. For k = 1,
i.e. C e eo is connected, (17) follows from the equation
lim P(Im(Y) n C = 0) = lim P(mY n C=0)m,
m-+oo
m-+oo
with
dm(Z) = dm({C}) = 1
and
qm(C) = 0.
Let c(C) = 0 and note that
mY n C = 0 4 mY n convC = 0.
As the inductionhypothesis,assumethat(17), (18), (19), and(20) hold for all C e Cowith at
most k connectedcomponents. Now considera set C e Co with k + 1 connectedcomponents.
The event
{Im(Y)nC = 0}
will be partitionedinto the disjointevents that, first,conv C is not intersectedby Y duringthe
firsti, i = 0,..., m, iterationsteps; second, in the (i + 1)th step C is separatedby exactly one
facet of Y into {D1, D2} e Z(C); and, third, D1 and D2 are not intersected in the remaining
m - i - 1 iterationsteps. Thereis still a 'residue',namely the event thatin the (i + 1)th step C
is separatedby more than one facet of Y. The probabilityof this latterevent will be denoted
by qm(C). Notice that, accordingto the definition,the last m - i - 1 iterationsteps with mY
have to be writtenas
Im-i-1
-m-i-1
Y
= --Im-i-1(Y).
m-i-1
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CrackSTITtessellations
SGSA
879
Denote by {D1 $ D2} the event that D1 and D2 are separatedby a facet of Y. The sets D1 and
D2 have at most k connectedcomponentsand, thus, the inductionhypothesiscan be appliedto
eitherof them. Define
m-l
P(mY n conv C = 0)'
qm(C)=
ZeZ(C) i=0
x P(0my((conv C)) > 2, mY n C = 0, mY generates partition Z)
P Im-i-1
x
Di EZ
Y
(m-i-1
Di =
0
< card(Z(C))m P(Amy((conv C)) > 2).
Then
P(Im (Y) nC = 0)
= P(mY n conv C = 0)m
m-1
3
+
P(mY
n
C)) = 1, D1 $ D2)
conv C = 0)i P(my((conv
{DI,D2}eZ(C) i=0
Im-i-
xP
(m
(-mm-i-1
Y
m
m-i-1))
PIm-i
nD1 =0)
Y ND2=0
+ qm(C)
= P(mY n conv C = 0)m
+
L
dm (Zl)dm (Z2)
{DI,D2}eZ(C) ZieZ(D1) Z2eZ(D2)
P(0my((conv C)) = 1, D1 $ D2)
P(mY n conv C = 0) - n5ezlUZ2 P(mY n conv S = 0)
x P(mY nconvC=
0)m -
P(mY n conv S = 0)m
SEZ1UZ2
+ qm (C)
(24)
if the denominatoris nonzero. By (15), there can be only finitely many m E N such that
the denominatorvanishes. A rearrangementin the sum in (24) yields the coefficients dm(Z),
Z e Z(C), and also makes theirproperty(18) obvious.
It remainsto consider qm(C) and to prove the properties(19) and (20). For j = 1, 2, the
inductionhypothesisyields, for Dj,
(m-i-1
Sc --
'
Dj (m - i - 1) P
(mi-1)y
con
Dj
< c(Dj)m P(myr((conv C)) > 2).
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2
W. NAGELAND V. WEISS
880 * SGSA
Note that
m-l
S
P(mY n conv C = 0)1 P( my((conv C)) = 1, D1 t D2)
{D1,D2}eZ(C)i=0
< m P(mY n conv C - 0)
< m.
Since
m-1'P(mY f conv C = 0)1i
S-
m and, from (17),
ZJ
m-i-1m
P(mY
Zj EZ(Dj)
n
convS
= 0)m-i-1
j = 1,2,
1,
Sj EZj
we obtain
m-l
qm(C) =
C
P(Amy((convC))
=
r[
P(mY nconvC
1, Di S D2)
{DI,D2}eZ(C)
= 0)i
i=0
i
d,
(m-i-1
d
x
1m--------2i
H
P(mY n conv S2 = 0)m-i-1qm-il-
D2m-
Z2 EZ(D)
( m5i€Z
dm-i--1
+ q-
((
P(m
xm82 E
i-
Z2
C))2
conv
= 0)m-i-lqm-i-
m
D
S2EZ2
D1
+qm-i-1
5
D2
qm-i-1
+qm(C)
m P(0my((conv C)) > 1, DI $ D2)
{Di, D2}eZ(C)
x [c(D2) + c(Di) + c(D1)c(D2)m P(my((conv
x m P(my((conv
C)) > 2)]
C)) > 2) + qm(C)
< c(C)m P(4my((conv C)) > 2),
with
c(C) = card(Z(C)) + max
mP(QFy((convC)) >
1, Di $ D2)
{D1,D2}EZ(C)
x [c(D2) + c(Di) + c(Di)c(D2)m
P(cmy((conv C)) > 2)].
(25)
The existence of such an upper bound c(C), depending on C and Y but not m, is due to
Lemma 6, which implies that both m P(Qmy((conv C)) > 2) and m P(cmy((conv C)) > 1)
have finite limits as m -+ oo. Equation (20) follows from (25) and the stationarity of Y. Finally,
note that (19) and Lemma 6 in fact imply (23).
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CrackSTITtessellations
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881
Proof of (ii) by induction over k. For k = 1, (21) is the assertion of Lemma 3, since
[C] = [conv C]. As the inductionhypothesis,assumethat(21) is truefor all those C e Co with
at most k connectedcomponents. Now considera set C e Cowith k + 1 connectedcomponents
and assume that (16) holds. Lemma4 with W D C and the inductionhypothesisyield
C = 0)
P(Y(a)
= e-aA([convC])
| D2])
aA([D
dte-taA([convC])
{D1,D2}eZ(C)
x
3
d(Z1)d(Z2)
C
Z1iZ(D1) Z2EZ(D2)
1
e-(1-t)aA([cnvS])
SEZ1UZ2
-aA([conv C])
SL
-A([D1
-A([conv C]) +
{D1,D2}EZ(C) Z1EZ(Di) Z2EZ(D2)
x (e-aA([cnvC]) - exp -a
I D2])
d(Z1)d(Z2)
A([conv S])
SEZ1UZ2
A([convS])l)
L
(26)
SEZ1UZ2
if all the denominatorsare nonzero,which is guaranteedby (16). A rearrangementof the items
yields (21).
Proof of (iii) by inductionover k. For k = 1, (22) is obvious since at the beginningof the
proof of (i) it was observedthatdm({C}) = 1, and from Lemma3 we have d({C}) = 1. As the
inductionhypothesis, assume that (22) is true for all those C E COwith at most k connected
components.Now considera set C e Cowith k + 1 connectedcomponentsandassumethat(16)
holds. We makeuse of the analogousstructuresof (24) and (26) andcomparecoefficients. The
inductionhypothesis yields limm-o dm(Zj) = d(Zj), j = 1, 2. It remainsto consider the
limit of the ratio. From
P(mY n conv S = 0) = 1 - P(mY n convyS
0),
we obtain
H
P(mYn convyS = 0)=l-
P(mYn convy
S
0)+r(m),
SEZIUZ2
SEZ1UZ2
wherer (m) is a sumof all termscontainingat least two factorsof the formP(mY n conv S = 0).
Thus, the version of Korolyuk'stheoremthatis given in Lemma 6 yields
_
P(Qmy((conv C)) = 1, D1 $ D2)
lim
lim
m-oo P(mY n conv C = 0) - 1SeZiuz2 P(mY n conv S = 0)
= lim
m-oo
m P(Pmy((conv C)) = 1, D1 $ D2)
-m P(mY 0 conv C # 0) +
sE Ziluz2m P(mY 0 conv S = 0) + o(1/m)
A([DlI D2])
-A([conv C]) + EezlZUZ2 A([conv S])
Obviously,(23) is a consequenceof Lemma 6 and (19).
This completes the proofs of (17) to (23), andthe assertionof the theoremfollows for a = 1
since, accordingto Lemma 7, we have
lim P(mY n conv S = 0)m = e-A([con S]).
m- oo
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W. NAGEL AND V. WEISS
882 *SGSA
As an immediate consequence of this theorem, we obtain the following uniquenessresult
and, thus, a characterizationof all stationarySTIT tessellations.
and directional
Corollary 2. A stationary random tessellation Y in Wd with 0 < Sv < 00oo
distribution,1 is STITif and only if
Y
Y(1),
where Y(1) is the crackSTITtessellation of Theorem1 with A = A(Sv,
, .).
7. Notes and concluding remarks
In the presentpaper,the notion of stationarityhas always been used in the sense of a spatial
homogeneity,i.e. the invarianceof the distributionwith respectto translationsin space. Another
possibility is to consider the repeatedrescaled iterationIm(Y) of a certaintessellation Y as a
randomprocess in discretetime m = 2, 3,.... Thenthe STITpropertyof the cracktessellation
Y(1) can be interpretedas stationarityof Im(Y(1)) with respectto time.
In an earlierpaper [11], the authorsstudied an approximationof the tessellation Y(a, W)
(for d = 2 anda particularclass of directionaldistributions)by a constructionof tessellationsin
which the exponentialtimes rj arereplacedby a geometricallydistributednumberof time steps.
The results given there supportthe assertionsof the presentpaperfrom anotherperspective.
In [10] it was shown that any planar STIT tessellation necessarily has a Poisson typical
cell and a.s. only 'T-shapednodes'. These propertiesof the crack STIT tessellations allow the
derivationof formulaefor severalmean values and distributionsof theirparameters.This will
be done in a forthcomingpaper [12]. Note that the Poisson typical cell propertycan also be
derivedfrom Lemma 3.
Concerningstereology,it is obvious that the STIT propertyof a tessellation is inheritedby
the tessellationsthatappearon lower-dimensionalplanarsections. Thus,the characterizationof
all stationarySTIT tessellationsgiven in Section 6 is a basis for the derivationof stereological
formulae.
Miles andMackisack[9], withreferenceto [2], were the firstto constructclasses of stationary
two-dimensionaltessellationswith Poisson typical cells andT-shapednodes only. One can see
thatthose tessellationsare not STIT;hence, the crackSTITtessellationsform a new class with
the propertiesmentioned.
In 1984, Cowan [3] presentedan idea for a tessellation that models the repeateddivision
of (biological) cells. He also proposed independentexponential lifetimes for the cells and
posed the problemof makingan appropriatechoice of parameters.Indeed,for the crack STIT
tessellations, the adjustmentof the parametersof the exponential distributionswas the key
problemto be solved.
There is a huge amount of literaturewith the keywords 'crack structures'or 'fracture
structures',anda varietyof models havebeen studied. Forseveralof these models, a theoretical
investigationis hardto perform.For some types of such structures,the crackSTITtessellations
can probablyserve as reasonableapproximatemodels, for examplein cases in which the growth
of the structurecanbe imaginedas a consecutivenestingof crackevents,withlatercracksending
at earlier cracks. This yields some hierarchicalstructure,as can be observed on surfaces of
potterysuch as Rakuor Majolika(referredto as the craqueldeeffect); for an image see [11].
It is a commonobservationin statisticsthatthe applicationof models with stabilityproperties
can be fruitful. This is well known for the Boolean model, which is stable with respect to the
union of sets, and probablyalso truefor crack STIT tessellations.
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Crack STIT tessellations
SGSA* 883
Acknowledgements
We are very gratefulto Joseph Mecke for stimulatingand helpful discussions. We thank
Joachim Ohser for fruitful remarksand for simulationsof the tessellations, in particularfor
Figure 1. Furthermore,we acknowledgethe helpful advice and suggestionsof the refereesand
of the CoordinatingEditor.
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