STIT Tessellations via Martingales

STIT Tessellations via Martingales Christoph Thäle Department of Mathematics, Osnabrück University (Germany) Stochastic Geometry Days, Lille, 30.3.-1.4.2011 Christoph Thäle (Uni Osnabrück) STIT Tessellations 1 / 62

This is part of joint work with Tomasz Schreiber from Toruń (25.6.1975-1.12.2010) Christoph Thäle (Uni Osnabrück) STIT Tessellations 2 / 62

Overview The model Martingales and STIT tessellations First applications Exact variance of the total surface area Exact variances for other intrinsic volumes Asymptotic expressions Limit theorems Christoph Thäle (Uni Osnabrück) STIT Tessellations 3 / 62

Construction and Key Properties We interpret a random tessellation as a special random closed set in R d. STIT tessellations Y (t) formally arise as limits of rescaled iterations of tessellations. In bounded windows W there is an explicit construction of Y (t, W ) in terms of waiting times. There is also an explicit global construction of STIT tessellations Y (t) (Poisson point process on a complicated state space). Important for later considerations: These constructions allow an interpretation of Y (t, W ) (or Y (t)) as a Markov process on (0, ) with values in the space of tessellations. Christoph Thäle (Uni Osnabrück) STIT Tessellations 4 / 62

Waiting Time Construction Let W R d be a compact convex polytope and Λ a non-degenerate locally finite hyperplane measure. Assign to W an exponentially distributed lifetime with parameter Λ([W ]). Upon expiry of this life time, a hyperplane is chosen according to Λ([W ]) 1 Λ( [W ]), is introduced in W and splits W into two polyhedral sub-cells W + and W. The construction is now continued recursively and independently in W + and W until some deterministic time threshold t > 0 is reached. Until time t > 0 there was constructed a random tessellation Y (t, W ) inside W with polyhedral cells. Christoph Thäle (Uni Osnabrück) STIT Tessellations 5 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 6 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 7 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 8 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 9 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 10 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 11 / 62

Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 12 / 62

Isotropic random STIT Tessellation in 2D Christoph Thäle (Uni Osnabrück) STIT Tessellations 13 / 62

Anisotropic random STIT Tessellation in 2D Christoph Thäle (Uni Osnabrück) STIT Tessellations 14 / 62

Isotropic random STIT Tessellation in 3D Christoph Thäle (Uni Osnabrück) STIT Tessellations 15 / 62

Anisotropic random STIT Tessellation in 3D Christoph Thäle (Uni Osnabrück) STIT Tessellations 16 / 62

Extension to R d We cannot start with a single hyperplane at time 0! But, Nagel und Weiß (2005) have shown that Y (t, W ) is spatially consistent, i.e for V W convex, we have Y (t, V ) = D Y (t, W ) V. Thus, by a consistency theorem there exists a random closed set Y (t) in R d with Y (t) W = D Y (t, W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 17 / 62

Parameter and special cases The law of Y (t) is characterized by t and Λ. In general, Y (t) is neither stationary nor isotropic. If Λ is translation-invariant, then Y (t) is stationary and thus a STIT tessellation. If Λ is the the isometry-invariant measure Λ iso, then Y (t) is stationary and isotropic. Christoph Thäle (Uni Osnabrück) STIT Tessellations 18 / 62

Generator of Y (t, W ) The random process Y (t, W ) is a pure jump Markov process on the space of tessellations in W and its generator L := L Λ;W is given by: LF (Y ) = [F (Y {f }) F (Y )] Λ(dH), [W ] }{{} f Cells(Y H) add-one cost with F measurable on the space of tessellation in W for which the integral is well defined. Christoph Thäle (Uni Osnabrück) STIT Tessellations 19 / 62

A derived Martingale From the standard theory (Dynkin s formula) it follows that the following real-valued process is a martingale with respect to the natural filtration of Y (t, W ): t F (Y (t, W )) LF (Y (s, W ))ds 0 for F D(L). Christoph Thäle (Uni Osnabrück) STIT Tessellations 20 / 62

A more special case Denote by MaxFacets(Y ) the set of maximal facets of the tessellation Y. We regard now the special function Σ φ (Y ) := φ(f ) f MaxFacets(Y ) with φ measurable and bounded on the space of (d 1)-polytopes in W. Applications of the general result from the last slide shows that t Σ φ (Y (t, W )) φ(f )Λ(dH)ds 0 [W ] is a martingale. f Cells(Y H) Christoph Thäle (Uni Osnabrück) STIT Tessellations 21 / 62

First Consequences Consider a STIT tessellation Y (t, W ) = Y (tλ, W ) and the measures M Y (t,w ) := δ c, M Y (t,w ) := EM Y (t,w ) c Cells(Y (t,w )) and likewise M PHT(tΛ,W ) and M PHT(tΛ,W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 22 / 62

First Consequences Consider a STIT tessellation Y (t, W ) = Y (tλ, W ) and the measures M Y (t,w ) := c Cells(Y (t,w )) δ c, M Y (t,w ) := EM Y (t,w ) and likewise M PHT(tΛ,W ) and M PHT(tΛ,W ). Define further F Y (t,w ) k := δ f, F Y (t,w ) k := EF Y (t,w ) k, k = 1,..., d 1, and F PHT(tΛ,W ) k := for k = 1,..., d 1,. f MaxFaces k (Y (t,w )) f Faces k (PHT(tΛ,W )) δ f, F PHT(tΛ,W ) k := EF PHT(tΛ,W ) k Christoph Thäle (Uni Osnabrück) STIT Tessellations 23 / 62

First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k 1 1 0 s FPHT(sΛ,W ) k ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 24 / 62

First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k 1 1 0 s FPHT(sΛ,W ) k This means that STITs and PHTs have the same typical cell distribution, but the spatial arrangement of the cells is different in both tessellation models. ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 25 / 62

First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. Idea of the proof: M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k 1 1 0 s FPHT(sΛ,W ) k (1) uses a uniqueness theorem for the solution of a certain operator differential equation on a space of measures together with a derived martingale. (2) uses a derived martingale, part (1), Slivnyak s theory for Poisson point process, scaling properties and adjacency relationships for Poisson hyperplane tessellations. ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 26 / 62

Application: Typical k-dimensional maximal faces Let us consider the isotropic case, i.e. Λ = Λ iso and Y (t) = Y (tλ iso ). Fix a measurable and translation-invariant ϕ k : MaxFaces k R, ϕ 0. Define the ϕ k -density of Y (t): ϕ k (Y (t)) = lim r 1 r d Vol d (W ) E Apply now the theory from above. f MaxFaces k (Y (t,rw )) ϕ k (f ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 27 / 62

Application: Typical k-dimensional maximal faces We have ϕ k (Y (t)) = (d k)2 d k 1 t For ϕ k 1 this yields 0 1 s ϕ k(pht(s))ds. t ( ) ( ) N k = (d k)2 d k 1 1 d d κd 1 κ d s d ds 0 s k dκ d = (d k)2 d k 1 κ ( ) ( ) d d d κd 1 t d. d k dκ d Christoph Thäle (Uni Osnabrück) STIT Tessellations 28 / 62

Application: Typical k-dimensional maximal faces Now the same procedure for ϕ k (f ) := 1[ ](f c(f )) with some center function c, f MaxFaces k (Y (t)). We get N k Q Y (t) k = (d k)2 d k 1 t 0 1 s NPHT(s) k Q PHT(s) k ds (the distribution of the naked polytopes). Inserting the values for N k and N PHT(s) k t Q Y (t) k = 0 ds d 1 t d gives Q PHT(s) k ds. The mixing distribution is a beta-distribution on (0, t) with parameters d and 1. Christoph Thäle (Uni Osnabrück) STIT Tessellations 29 / 62

Application: Typical maximal segment Let p (d) l (x) be the length density of the typical maximal Segment in R d. Then p (d) l (x) = t 0 λ 1 se λ 1sx dsd 1 t d ds = Γ ( ) d 2 λ 1 = ( πγ d+1 ). 2 d (λ 1 t) d x d+1 γ(d + 1, λ 1tx) with 1 ( d = 2: t 2 x 3 π 2 (π 2 + 2πtx + 2t 2 x 2 )e 2 tx) π, mean π t, no higher moments exist. 3 ( d = 3: t 3 x 4 48 (48 + 24tx + 6t 2 x 2 + t 3 x 3 )e 1 tx) 2, mean 3 t, variance 24, no higher moments exist. t 2 Christoph Thäle (Uni Osnabrück) STIT Tessellations 30 / 62

Application: Typical k-dimensional maximal faces For the mean j-th intrinsic volume of the typical k-dimensional maximal face, 0 j k, we have d EV j (I k ) = (d j)κ j ( k j ) ( 2 πγ ( ) d+1 2 Γ ( ) d 2 ) j 1 t j. Christoph Thäle (Uni Osnabrück) STIT Tessellations 31 / 62

Next goal The next goal is to use the martingale approach to study second-order properties of the STIT tessellations, such as variances and covariances. Christoph Thäle (Uni Osnabrück) STIT Tessellations 32 / 62

Another derived Martingale - Notation Let φ be bounded and measurable on the space of (d 1)-polytopes. Define Σ φ (Y (t, W )) := φ(f ) and A φ (Y (t, W )) := f MaxFacets(Y (t,w )) [W ] f Cells(Y (t,w ) H) φ(f )Λ(dH) and put Σ φ (Y (t, W )) := Σ φ (Y (t, W )) EΣ φ (Y (t, W )), Ā φ (Y (t, W )) := A φ (t, W ) EA φ (t, W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 33 / 62

Another derived Martingale Let φ be bounded and measurable on the space of (d 1)-polytopes. Define and Σ φ (Y (t, W )) := A φ (Y (t, W )) := A second-order martingale The random process Σ 2 φ (Y (t, W )) t 0 f MaxFacets(Y (t,w )) [W ] f Cells(Y (t,w ) H) φ(f ) φ(f )Λ(dH). t A φ 2(Y (s, W ))ds 2 Ā φ (Y (s, W )) Σ φ (Y (s, W ))ds 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Christoph Thäle (Uni Osnabrück) STIT Tessellations 34 / 62

Variance of the Total Surface Area A second-order martingale The random process Σ 2 φ (Y (t, W )) t 0 t A φ 2(Y (s, W ))ds 2 Ā φ (Y (s, W )) Σ φ (Y (s, W ))ds 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Take φ(f ) := Vol d 1 (f ). Then, A Vold 1 (Y (, W )) is constant, thus Ā Vold 1 0. Taking expectations in the above formula, we get Var Vol d 1 (Y (t, W )) = t It remains to calculate EA Vol 2 d 1 (Y (s, W )). 0 EA Vol 2 d 1 (Y (s, W ))ds. Christoph Thäle (Uni Osnabrück) STIT Tessellations 35 / 62

Calculation of EA Vol 2 d 1 (Y (s, W )) EA Vol 2 (Y (s, W )) d 1 = E = E = [W ] f Cells(Y (s,w ) H) [W ] [W ] W H W H W H W H Vol 2 d 1 (f )Λ(dH) 1[x, y in the same cell of Y (s, W ) H]dxdyΛ(dH) e sλ([xy]) dxdyλ(dh), since STIT tessellations have Poisson typical cells and Y (s) H is also a STIT tessellation. Christoph Thäle (Uni Osnabrück) STIT Tessellations 36 / 62

Variance of the Total Surface Area Putting together and Var Vol d 1 (Y (t, W )) = EA Vol 2 (Y (s, W )) = d 1 [W ] t 0 EA Vol 2 d 1 (Y (s, W ))ds W H W H e sλ([xy]) dxdyλ(dh), we get by integration Var Vol d 1 (Y (t, W )) = [W ] W H W H 1 e tλ([xy]) dxdyλ(dh). Λ([xy]) Christoph Thäle (Uni Osnabrück) STIT Tessellations 37 / 62

Variance of the Total Surface Area Assume that Λ = Λ iso. Then, the affine Blaschke-Petkantschin formula implies g(x, y)dxdyλ iso (dh) Take now [W ] W H W H = (d 1)κ d 1 dκ d W W g(x, y) x y dxdy. g(x, y) = 1 exp( tλ iso([xy])) Λ iso ([xy]) = 1 e 2κ d 1 dκ d 2κ d 1 t x y dκ d x y. Christoph Thäle (Uni Osnabrück) STIT Tessellations 38 / 62

Variance of the Total Surface Area Theorem (Schreiber and T. 2010) For a stationary and isotropic STIT tessellation Y (t) we have Var(Vol d 1 (Y (t, W ))) = d 1 1 e 2κ d 1 t x y dκ d 2 W W x y 2 dxdy = d(d 1)κ ( ) d γ 2 W (r)r d 3 1 e 2κ d 1 tr dκ d dr. 0 Christoph Thäle (Uni Osnabrück) STIT Tessellations 39 / 62

Variance of the Total Surface Area Theorem (Schreiber and T. 2010) For a stationary and isotropic STIT tessellation Y (t) we have Var(Vol d 1 (Y (t, W ))) = d(d 1)κ ( ) d γ 2 W (r)r d 3 1 e 2κ d 1 tr dκ d dr. Corollary (Weiß, Ohser, Nagel, d = 2 and Schreiber and T. in general) The pair-correlation function g d (r) of the random surface measure of a stationary and isotropic Y (t) is given by g d (r) = 1 + d 1 ( ) 2t 2 r 2 1 e 2κ d 1 tr dκ d. 0 Christoph Thäle (Uni Osnabrück) STIT Tessellations 40 / 62

Variances for Intrinsic Volumes We consider only the stationary and isotropic case Λ = Λ iso. Our aim is to calculate Var Σ Vj (Y (t, W )), where, recall, Σ Vj (Y (t, W )) = V j (f ). f MaxFacets(Y (t,w )) However, we start with F j (Y (t, W )) := V j (c). c Cells(Y (t,w )) If f splits c into c + and c, we have F j (Y (t, W ) {f }) F j (Y (t, W )) = V j (c + )+V j (c ) V j (c) = V j (f ), thus F j (Y (t, W )) = Σ Vj (Y (t, W )) + V j (W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 42 / 62

Variances for Intrinsic Volumes Recall, that A Vj (Y (t, W )) = [W ] f Cells(Y (t,w ) H) V j (f )Λ iso (dh). Crofton s formula and F j (Y (t, W )) = Σ Vj (Y (t, W )) + V j (W ) implies now A Vj (Y (t, W )) = V j (c H)Λ iso (dh) = c Cells(Y (t,w )) c Cells(Y (t,w )) [W ] γ j+1 V j+1 (c) = γ j+1 F j+1 (Y (t, W )) = γ j+1 (Σ Vj+1 (Y (t, W )) + V j+1 (W )), thus Ā Vj (Y (t, W )) = γ j+1 Σ Vj+1 (Y (t,w )). Christoph Thäle (Uni Osnabrück) STIT Tessellations 43 / 62

Another Martingale Denote Σ φ;t := Σ φ (Y (t, W )). By considering once again the time-augmented process (Y (t, W ), t) and Dynkin s formula one shows that t t Σ Vi ;t Σ Vj ;t A Vi V j ;sds [Ā Vi ;s Σ Vj ;s + Ā Vj ;s Σ Vi ;s]ds 0 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Using that Ā Vj (Y (t, W )) = γ j+1 Σ Vj+1 (Y (t,w )), also t Σ Vi ;t Σ Vj ;t A Vi V j ;sds 0 is a martingale as well. t 0 [γ i+1 ΣVi+1 ;s Σ Vj ;s + γ j+1 ΣVi ;s Σ Vj+1 ;s]ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 45 / 62

The Recursion Formula Using that ĀV j (Y (t, W )) = γ j+1 ΣVj+1 (Y (t,w )), also t Σ Vi ;t Σ Vj ;t A Vi V j ;sds 0 is a martingale as well. t Taking expectation, we get the recursion formula + t 0 Cov(Σ Vi ;t, Σ Vj ;t) = 0 [γ i+1 ΣVi+1 ;s Σ Vj ;s + γ j+1 ΣVi ;s Σ Vj+1 ;s]ds t 0 EA Vi V j ;sds [γ i+1 Cov(Σ Vi+1 ;s, Σ Vj ;s) + γ j+1 Cov(Σ Vi ;s, Σ Vj+1 ;s)]ds. The recursion terminates, since Σ Vd ;t 0, which allows an explicit expression for all Cov(V i;t, V j;t ). Recall that Var Σ Vd 1 ;t = t 0 EA V 2 d 1 ;sds. Christoph Thäle (Uni Osnabrück) STIT Tessellations 46 / 62

Exact Expression I n (f ; t) := = t s1 0 0 1 (n 1)! sn 1 0 t Theorem (Schreiber and T. 2010) 0 f (s n )ds n ds n 1 ds 1 (t s) n 1 f (s)ds. The covariance between Σ Vd 1 k ;t and Σ Vd 1 l ;t for k, l {0,..., d 1} of a stationary and isotropic random STIT tessellation Y (t, W ) is given by ( k i=m+1 Cov(Σ Vd 1 k ;t, Σ Vd 1 l ;t) = γ d i ) l j=n+1 γ d j k m=0 n=0 l ( ) k + l m n k m I k+l m n+1 (EA Vd 1 m V d 1 n ;( ); t). Christoph Thäle (Uni Osnabrück) STIT Tessellations 47 / 62

Asyptotic Expressions: The Problem Take a convex body W R d and consider the sequence W R := R W as R. How does Var Σ Vi (Y (t, W R )) behave, as R? Which terms dominate the expression? How is the formula influenced by the geometry of W? The answers for d = 2 and d 3 are different! For simplicity we consider only d 3. Christoph Thäle (Uni Osnabrück) STIT Tessellations 48 / 62

Asymptotics Proposition For k, l, m, n {0,..., d 1}, with t fixed we have with p := k + l m n + 1 I p (EA W R V d 1 m V d 1 n ;( ) ; t) = O(R 2(d 1) m n ), if m + n d 3, O(R d log R), if m + n = d 2, O(R 2d 1 m n ), if m + n d 1. Christoph Thäle (Uni Osnabrück) STIT Tessellations 49 / 62

Asymptotic picture Proposition For k, l, m, n {0,..., d 1}, with t fixed we have with p := k + l m n + 1 I p (EA W R V d 1 m V d 1 n ;( ) ; t) = Corollary O(R 2(d 1) m n ), if m + n d 3, O(R d log R), if m + n = d 2, O(R 2d 1 m n ), if m + n d 1. The asymptotic covariance Cov(Σ Vd 1 k ;t, Σ Vd 1 l ;t) is dominated by the term n = m = 0, i.e. by I k+l+1 (EA W R ; t) = I k+l (Var Σ W Vol 2 R V d 1 d 1 ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 50 / 62

Influence of W The geometry of W is for d 3 reflected by either one of the (non-additive) functionals dxdy E 2 (W ) = x y 2 or since I d 1 (W ) = dκ d 2 E 2 (W ) = W L W Vol d 1 1 (W L)dL, 2 (d 1)(d 2) I d 1(W ). For d = 2 only the area Vol 2 (W ) matters, but not the precise shape of W. But here some logarithm enters the discussion... Christoph Thäle (Uni Osnabrück) STIT Tessellations 51 / 62

The Asymptotic result Theorem (Schreiber and T. 2010) Asymptotically, as R, we have for k, l {0,..., d 1} ( k ) 1 γ d i d 2 i=1 Cov(Σ W R V d 1 k ;t, ΣW R V d 1 l ;t ) = l j=1 γ d j tk+l k!l! I d 1(W )R 2(d 1) + O(R 2d 3 ) and for k = l ( k ) 2 Var(Σ W R V d 1 k ;t ) = 1 t 2k γ d i d 2 (k!) 2 I d 1(W )R 2(d 1) + O(R 2d 3 ). i=1 Christoph Thäle (Uni Osnabrück) STIT Tessellations 52 / 62

Limit theorems We restrict ourself from now on to the case of the total surface area, but limit theorems for other functionals are also available. We approach the problem in two different (but closely related) settings: We can consider the surface area constructed by facets born after an initial time instant s 0 > 0. It is also natural to ask for the surface area constructed by all facets. Interestingly both approaches lead to results of very different qualitative nature! Christoph Thäle (Uni Osnabrück) STIT Tessellations 53 / 62

First limit theorem Theorem (Schreiber and T. 2010) For each s 0 > 0, the random variable 1 R d/2 [(Vol d 1(Y (1, W R )) EVol d 1 (Y (1, W R ))) (Vol d 1 (Y (s 0, W R )) EVol d 1 (Y (s 0, W R )))] converges, as R, in law to N (0, V W (Vol d 1, Λ) 1 s 0 s 1 d ds), a normal distribution with mean 0 and variance V W (Vol d 1, Λ) 1 s 0 s 1 d ds, where V W (Vol d 1, Λ) is explicitly known. Christoph Thäle (Uni Osnabrück) STIT Tessellations 54 / 62

First limit theorem Theorem (Schreiber and T. 2010) For each s 0 > 0, the random variable 1 R d/2 [(Vol d 1(Y (1, W R )) EVol d 1 (Y (1, W R ))) (Vol d 1 (Y (s 0, W R )) EVol d 1 (Y (s 0, W R )))] converges, as R, in law to N (0, V W (Vol d 1, Λ) 1 s 0 s 1 d ds), a normal distribution with mean 0 and variance V W (Vol d 1, Λ) 1 s 0 s 1 d ds, where V W (Vol d 1, Λ) is explicitly known. Idea of the proof: Use STIT scaling and ergodicity to calculate the asymptotic variance. Check the assumptions of the martingale CLT for cadlag martingales. Christoph Thäle (Uni Osnabrück) STIT Tessellations 55 / 62

Planar CLT Theorem (Schreiber and T. 2010) We have for the stationary and isotropic STIT tessellation Y (1) in the plane 1 R log R [Vol 1(Y (1, W R )) EVol 1 (Y (1, W R ))] = N (0, π Vol 2 (W )), where = means convergence in law. Idea of the proof: Use a space-time scaling to construct an asymptotically equivalent martingale with good properties. Check again the assumptions of the martingale CLT for cadlag martingales. Christoph Thäle (Uni Osnabrück) STIT Tessellations 57 / 62

Higher dimensions Theorem (Schreiber and T. 2010) For d > 2, Λ = d + i=1 δ re i +ei dr and W = [0, 1] d, R 2(d 1) [Vol d 1 (1, W R ) EVol d 1 (1, W R )] converges, as R, to a non-gaussian square-integrable random variable Ξ(W ) with explicitly known variance. Idea of the proof: Show by a conditioning argument that Ξ(W ) satisfies P(Ξ(W ) > ξ) = exp( Θ(ξ log ξ)), whereas a Gaussian random variable has tails of order exp( Θ(ξ 2 )). Christoph Thäle (Uni Osnabrück) STIT Tessellations 59 / 62

Why this difference? Overall variance is of order R 2(d 1) for d > 2 and of order R 2 log R for d = 2. The variance contribution after the initial time instant s 0 > 0 is of order R d for all d. Thus, for d = 2 the initial segments contribute a negligible amount to the total variance. But, in higher dimensions d > 2, the big bang phase is crucial and dominates the scenery. Christoph Thäle (Uni Osnabrück) STIT Tessellations 60 / 62

Thank you for your attention! Christoph Thäle (Uni Osnabrück) STIT Tessellations 61 / 62