G¨ unter Last Institut f¨ ur Stochastik Universit¨ at Karlsruhe (TH) Distributional properties of Poisson Voronoi tessellations G¨ unter Last Universit¨ at Karlsruhe (TH) joint work with Volker Baumstark (Karlsruhe) Prague Stochastics 2006 Charles University 22.08.2006
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Gunter Last
Institut fur Stochastik
Universitat Karlsruhe (TH)
Distributional properties of Poisson Voronoi tessellations
Gunter Last
Universitat Karlsruhe (TH)
joint work with Volker Baumstark (Karlsruhe)
Prague Stochastics 2006
Charles University
22.08.2006
Gunter Last Distributional properties of Poisson Voronoi tessellations
1. Voronoi tessellations
Definition:
(i) The space of all point configurations in Rd is defined as
N := {ϕ ⊂ Rd : ϕ is locally finite}.
(ii) Any ϕ ∈ N is identified with a counting measure:
ϕ(B) := card{x ∈ ϕ : x ∈ B}, B ⊂ Rd.
(iii) The σ-field N is the smallest σ-field of subsets of N making the
mappings ϕ 7→ ϕ(B) for all Borel sets B ⊂ Rd measurable.
22.08.2006, Prague Stochastics Slide 2/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: The points of ϕ ∈ N are in general quadratic position if
the following two conditions are satisfied.
(i) Any k ∈ {2, . . . , d+ 2} points of ϕ are in general position.
(ii) No d+ 2 points of ϕ lie on the boundary of some ball.
22.08.2006, Prague Stochastics Slide 3/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let ϕ ∈ N.
(i) The Voronoi cell C(ϕ, x) of x ∈ ϕ is the set of all sites y ∈ Rdwhose distance from x is smaller or equal than the distances to
all other points of ϕ.
(ii) The Voronoi tessellation based on ϕ is the system
Sd(ϕ) := {C(ϕ, x) : x ∈ ϕ}.
Remark: If the convex hull of ϕ coincides with Rd, then all Voronoi
cells are bounded and Sd(ϕ) is a face-to face tessellation. Moreover,
if the point of ϕ are in general quadratic position, then Sd(ϕ) is also
normal in the sense that any k-face is contained in exactly d−k+ 1
cells.
22.08.2006, Prague Stochastics Slide 4/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
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22.08.2006, Prague Stochastics Slide 5/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let C be a convex polytope. Then
C =⋃
k∈{0,...,d}
⋃
C∈Sk(C)
relintF,
where Sk(C) is a finite set of k-dimensional polytopes whose affine
hulls are pairwise not equal. A polytope F ∈ Sk(C) is called a
k-face of C.
Definition: Let ϕ ∈ N and k ∈ {0, . . . , d}. The system of all k-faces
of the Voronoi tessellation Sd(ϕ) is defined by
Sk(ϕ) :=⋃
C∈Sd(ϕ)
Sk(C).
22.08.2006, Prague Stochastics Slide 6/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
2. Stationary point processes and random measures
Definition:
(i) For any x ∈ Rd the shift θx : N→ N is defined by
θxϕ = ϕ− x.
(ii) A probability measure P on (N,N ) is stationary if
P ◦ θx = P, x ∈ Rd.
22.08.2006, Prague Stochastics Slide 7/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Assumption: P is a stationary probability measure on (N,N ).
Definition:
(i) M denotes the space of all locally finite measures on Rd.
(ii) The σ-field M is the smallest σ-field of subsets of M making
the mappings α 7→ α(B) for all Borel sets B ⊂ Rd measurable.
(iii) A random measure M is a measurable mapping from N to M.
(iv) A random measure M is stationary if
M(ϕ,B + x) = M(θxϕ,B), ϕ ∈M, x ∈ Rd, B ∈ Bd.
Remark: The identity N on N is a stationary random measure.
22.08.2006, Prague Stochastics Slide 8/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let M be a stationary random measure.
(i) The intensity of M is the number
λM := E[M([0, 1]d)].
(ii) If λM is positive and finite, then
P0M (A) :=
1
λME[∫
1{θxN ∈ A, x ∈ [0, 1]d}M(dx)
], A ∈ N ,
is called Palm probability measure of M .
22.08.2006, Prague Stochastics Slide 9/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
3. Typical faces
Assumption: P is a stationary probability measure on (N,N ) such
that almost all ϕ ∈ N are non-empty and the points of almost all
ϕ ∈ N are in general quadratic position. We consider the (random)
Voronoi tessellation
Sd(N) = {C(N, x) : x ∈ N}
generated by N .
22.08.2006, Prague Stochastics Slide 10/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let k ∈ {0, . . . , d} and F ∈ Sk. Take some y in the
relative interior of F and assume that the points of N are in gen-
eral position. Then there are exactly d − k + 1 different points
X0, . . . , Xd−k ∈ N (the neighbours of F ) such that the distances
Ry := ‖Xi − y‖ are the same for all i and such that the open ball
with centre y and radius Ry does not contain any point of N . Let
πk(F ) denote the centre of the unique (d − k)-dimensional ball in
the affine hull of the neighbours containing the neighbours on its
boundary. Define the stationary point process of centres of k-faces
by
Nk := {πk(F ) : F ∈ Sk(N)}.
22.08.2006, Prague Stochastics Slide 11/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Assumption: For any k ∈ {0, . . . , d} the intensity
λk := E[Nk([0, 1]d)]
is assumed to be finite.
Remark: We have a.s. that N = Nd and hence λd = λ.
Definition: Let k ∈ {0, . . . , d}. Under the Palm probability measure
P0Nk
we denote by Ck ∈ Sk(N) the k-face satisfying π(Ck) = 0. The
distribution
P0Nk
(Ck ∈ ·)is the distribution of the typical k-face of the Voronoi tessellation
based on N .
22.08.2006, Prague Stochastics Slide 12/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: For any k ∈ {0, . . . , d} we define the stationary random
measure
Mk :=∑
F∈Sk(N)
Hk(F ∩ ·),
where Hk denotes k-dimensional Hausdorff measure on Rd.
Assumption: For any k ∈ {0, . . . , d} the intensity
µk := E[Mk([0, 1]d)]
is assumed to be finite.
Remark: We have M0 = N0 and hence λ0 = µ0.
22.08.2006, Prague Stochastics Slide 13/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let k ∈ {0, . . . , d}. Under the Palm probability measure
P0Mk
we denote by Fk ∈ Sk(N) the k-face satisfying 0 ∈ Fk. The
distribution
P0Mk
(Fk ∈ ·)can be interpreted as an area-biased version of the distribution of
the typical k-face.
Proposition: Consider k ∈ {0, . . . , d} and a measurable and shift-
invariant function g : N→ [0,∞). Then
µkE0Mk
[g] = λkE0Nk
[Hk(Ck) · g
],
λkE0Nk
[g] = µkE0Mk
[Hk(Fk)−1 · g
].
22.08.2006, Prague Stochastics Slide 14/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
4. Mean values for typical faces
Corollary: For any k ∈ {0, . . . , d} we have
µk = λkE0Nk
[Hk(Ck)
],
λk = µkE0Mk
[Hk(Fk)−1
].
In particular
E0Nd
[Hd(Cd)] = λ−1,
E[Hd(Fd)−1] = λ.
Proposition: We have
d∑
j=0
(−1)jλj = 0.
22.08.2006, Prague Stochastics Slide 15/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Definition: Let Pd denote the system of all convex polytopes in Rd.For k ∈ {0, . . . , d} we define νk : Pd → N by
νk(F ) := cardSk(F ).
Proposition: Consider the planar case d = 2. Then λ0 = 2λ and
λ1 = 3λ. Moreover,
E0N2
[H2(C2)] =1
λ,
E0N2
[H1(∂C2)] =2µ1
λ,
E0N2
[ν0(C2)] = 6,
E0N1
[H1(C1)] =µ1
3λ.
22.08.2006, Prague Stochastics Slide 16/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Theorem: If N is a stationary Poisson process of intensity λ then
the intensities µk are explicitly known. In case d = 2 we have
µ0 = 2λ, µ1 = 2√λ
and in case d = 3 we have
µ0 =24π2
35λ, µ1 =
48π2
35λ, µ2 =
(24π2
35+ 1)λ.
Problem: Assume thatN is a stationary Poisson process. Determine
the distributions
P0Nk
(Ck ∈ ·), k = 0, . . . , d,
and
P0Mk
(Fk ∈ ·), k = 0, . . . , d.
22.08.2006, Prague Stochastics Slide 17/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
5. The neighbours of a typical vertex
Assumption: N is a stationary Poisson process of intensity λ > 0.
Definition: Consider the probability measure P0N0
.
(i) Almost surely there are exactly d + 1 different points
X0, . . . , Xd ∈ N (lexicographically ordered) such that
{0} = C(N,X0) ∩ · · · ∩ C(N,Xd).
The points X0, . . . , Xd are the neighbours of the origin.
(ii) Let R := |X0| = · · · = |Xd| denote the distance to the neigh-
bours and define the unit vectors
U0 :=X0
R, . . . , Ud :=
Xd
R.
22.08.2006, Prague Stochastics Slide 18/ 29
Gunter Last Distributional properties of Poisson Voronoi tessellations
Theorem: Under the probability measure P0N0
the following holds.
(i) The random variables ({x ∈ N : |x| > R}, R) and (U0, . . . , Ud)
are independent.
(ii) Rd is Gamma distributed with shape parameter d and scale pa-
rameter γκd.
(iii) The conditional distribution of {x ∈ N : |x| > R} given R = r
is the distribution of a Poisson process restricted to the comple-