The extremal index for a random tessellation

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14258
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_19You do not have permission to access embedded form.

Résumé

Let m be a random tessellation in R d , d ≥ 1, observed in the window W p = ρ1/d[0, 1] d , ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus W p as ρ goes to infinity.When the normalized maximum converges, we show that its asymptotic distribution depends on the so-called extremal index. Two examples of extremal indices are provided for Poisson-Voronoi and Poisson-Delaunay tessellations.

The extremal index for a random tessellation

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Let m be a random tessellation in R d , d ≥ 1, observed in the window W p = ρ1/d[0, 1] d , ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus W p as ρ goes to infinity.When the normalized maximum converges, we show that its asymptotic distribution depends on the so-called extremal index. Two examples of extremal indices are provided for Poisson-Voronoi and Poisson-Delaunay tessellations.

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Random tessellations Main problem Extremal index The extremal index for a random tessellation Nicolas Chenavier Université Littoral Côte d’Opale October 28, 2015 Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Plan 1 Random tessellations 2 Main problem 3 Extremal index Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Random tessellations Definition A (convex) random tessellation m in Rd is a partition of the Euclidean space into random polytopes (called cells). We will only consider the particular case where m is a : I Poisson-Voronoi tessellation ; I Poisson-Delaunay tessellation. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Poisson-Voronoi tessellation I X, Poisson point process in Rd ; I ∀x ∈ X, CX(x) := {y ∈ Rd , |y − x| ≤ |y − x0 |, x0 ∈ X} (Voronoi cell with nucleus x) ; I mPVT := {CX(x), x ∈ X}, Poisson-Voronoi tessellation ; I ∀CX(x) ∈ mPVT , we let z(CX(x)) := x. x CX(x) Mosaique de Poisson-Voronoi Figure: Poisson-Voronoi tessellation. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Poisson-Delaunay tessellation I X, Poisson point process in Rd ; I ∀x, x0 ∈ X, x and x0 define an edge if CX(x) ∩ CX(x0 ) 6= ∅ ; I mPDT , Poisson-Delaunay tessellation ; I ∀C ∈ mPDT , we let z(C) as the circumcenter of C. x x0 z(C) Mosaique de Poisson-Delaunay Figure: Poisson-Delaunay tessellation. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Typical cell Definition Let m be a stationary random tessellation. The typical cell of m is a random polytope C in Rd which distribution given as follows : for each bounded translation-invariant function g : {polytopes} → R, we have E [g(C)] := 1 N(B) E     X C∈m, z(C)∈B g(C)     , where : I B ⊂ R is any Borel subset with finite and non-empty volume ; I N(B) is the mean number of cells with nucleus in B. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index 1 Random tessellations 2 Main problem 3 Extremal index Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Main problem Framework : I m = mPVT , mPDT ; I Wρ := [0, ρ]d , with ρ > 0 ; I g : {polytopes} → R, geometrical characteristic. Aim : asymptotic behaviour, when ρ → ∞, of Mg,ρ = max C∈m, z(C)∈Wρ g(C)? Figure: Voronoi cell maximizing the area in the square. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Objective and applications I Objective : find ag,ρ > 0, bg,ρ ∈ R s.t. P  Mg,ρ ≤ ag,ρt + bg,ρ  converges, as ρ → ∞, for each t ∈ R. I Applications : regularity of the tessellation ; discrimination of point processes and tessellations ; Poisson-Voronoi approximation. Approximation de Poisson-Voronoi Figure: Poisson-Voronoi approximation. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Asymptotics under a local correlation condition Notation : let vρ := ag,ρt + bρ be a threshold such that ρd · P (g(C) > vρ) −→ ρ→∞ τ, for some τ := τ(t) ≥ 0. Local Correlation Condition (LCC) ρd (log ρ)d · E      X (C1,C2)6=∈m2, z(C1),z(C2)∈[0,log ρ]d 1g(C1)>vρ,g(C2)>vρ      −→ ρ→∞ 0. Theorem Under (LCC), we have : P (Mg,ρ ≤ vρ) −→ ρ→∞ e−τ . Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index 1 Random tessellations 2 Main problem 3 Extremal index Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Definition of the extremal index Proposition Assume that for all τ ≥ 0, there exists a threshold v (τ) ρ depending on ρ such that ρd · P(g(C) > v (τ) ρ ) −→ ρ→∞ τ. Then there exists θ ∈ [0, 1] such that, for all τ ≥ 0, lim ρ→∞ P(Mg,ρ ≤ v(τ) ρ ) = e−θτ , provided that the limit exists. Definition According to Leadbetter, we say that θ ∈ [0, 1] is the extremal index if, for each τ ≥ 0, we have : ρd · P  g(C) > v(τ) ρ  −→ ρ→∞ τ and lim ρ→∞ P(Mg,ρ ≤ v(τ) ρ ) = e−θτ . Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Example 1 Framework : I m := mPVT : Poisson-Voronoi tessellation ; I g(C) := r(C) : inradius of any cell C := CX(x) with x ∈ X, i.e. r(C) := r (CX(x)) := max{r ∈ R+ : B(x, r) ⊂ CX(x)}. I rmin,PVT (ρ) := minx∈X∩Wρ r (CX(x)). Extremal index : θ = 1/2 for each d ≥ 1. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Minimum of inradius for a Poisson-Voronoi tessellation (b) Typical Poisson−Voronoï cell with a small inradii x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Example 2 Framework : I m := mPDT : Poisson-Delaunay tessellation ; I g(C) := R(C) : circumradius of any cell C, i.e. R(C) := min{r ∈ R+ : B(x, r) ⊃ C}. I Rmax,PDT (ρ) := maxC∈mPDT :z(C)∈Wρ R(C). Extremal index : θ = 1; 1/2; 35/128 for d = 1; 2; 3. Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Maximum of circumradius for a Poisson-Delaunay tessellation (d) Typical Poisson−Delaunay cell with a large circumradii x y −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 Nicolas Chenavier The extremal index for a random tessellation Random tessellations Main problem Extremal index Work in progress Joint work with C. Robert (ISFA, Lyon 1) : I new characterization of the extremal index (not based on classical block and run estimators appearing in the classical Extreme Value Theory) ; I simulation and estimation for the extremal index and cluster size distribution (for Poisson-Voronoi and Poisson-Delaunay tessellations). Nicolas Chenavier The extremal index for a random tessellation