Central limit theorems for random tessellations and random graphs - PowerPoint PPT Presentation

Central limit theorems for random tessellations and random graphs Matthias Schulte www.kit.edu KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

Poisson process in [ 0 , 1 ] d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ( X i ) 1 ≤ i ≤ M with independent X 1 , X 2 , . . . ∼ Uniform ([ 0 , 1 ] d ) and M ∼ Poisson ( t ) , t ≥ 0, i.e. P ( M = k ) = t k k ! e − t , k ∈ N ∪ { 0 } . June 25, 2015 2/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process in [ 0 , 1 ] d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ( X i ) 1 ≤ i ≤ M with independent X 1 , X 2 , . . . ∼ Uniform ([ 0 , 1 ] d ) and M ∼ Poisson ( t ) , t ≥ 0, i.e. P ( M = k ) = t k k ! e − t , k ∈ N ∪ { 0 } . Define η = � M i = 1 δ X i , where δ x is the Dirac measure at x ∈ R d , i.e., η ( A ) is the number of points of ( X i ) 1 ≤ i ≤ M in A ∈ B ( R d ) . June 25, 2015 2/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process in [ 0 , 1 ] d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Observe that η ( A 1 ) , . . . , η ( A n ) independent for disjoint A 1 , . . . , A n ∈ B ( R d ) June 25, 2015 2/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process in [ 0 , 1 ] d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Observe that η ( A 1 ) , . . . , η ( A n ) independent for disjoint A 1 , . . . , A n ∈ B ( R d ) η ( A ) ∼ Poisson ( t Vol ( A ∩ [ 0 , 1 ] d )) , A ∈ B ( R d ) June 25, 2015 2/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process Definition: A random counting measure η on a measurable space ( X , X ) is a Poisson process with σ -finite intensity measure λ if η ( A 1 ) , . . . , η ( A n ) are independent for all disjoint sets A 1 , . . . , A n ∈ X , n ∈ N , η ( A ) is Poisson distributed with parameter λ ( A ) for all A ∈ X . June 25, 2015 3/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process Definition: A random counting measure η on a measurable space ( X , X ) is a Poisson process with σ -finite intensity measure λ if η ( A 1 ) , . . . , η ( A n ) are independent for all disjoint sets A 1 , . . . , A n ∈ X , n ∈ N , η ( A ) is Poisson distributed with parameter λ ( A ) for all A ∈ X . In the following we identify η with its support and think of it as a random configuration of points. June 25, 2015 3/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson process Definition: A random counting measure η on a measurable space ( X , X ) is a Poisson process with σ -finite intensity measure λ if η ( A 1 ) , . . . , η ( A n ) are independent for all disjoint sets A 1 , . . . , A n ∈ X , n ∈ N , η ( A ) is Poisson distributed with parameter λ ( A ) for all A ∈ X . In the following we identify η with its support and think of it as a random configuration of points. Example: X = R d , λ = t Vol, t ≥ 0: stationary Poisson process of intensity t in R d June 25, 2015 3/22 M. Schulte – Central limit theorems for random tessellations and random graphs

k-Nearest Neighbour Graph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● June 25, 2015 4/22 M. Schulte – Central limit theorems for random tessellations and random graphs

k-Nearest Neighbour Graph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● June 25, 2015 4/22 M. Schulte – Central limit theorems for random tessellations and random graphs

k-Nearest Neighbour Graph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● What is the edge length of the k -nearest neighbour graph? June 25, 2015 4/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson-Voronoi tessellation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● June 25, 2015 5/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson-Voronoi tessellation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● June 25, 2015 5/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Poisson-Voronoi tessellation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● What is the edge length of the Poisson-Voronoi tessellation within the observation window? June 25, 2015 5/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Classical central limit Theorem Theorem: 1 < ∞ , let S n = � n Let ( Y i ) i ∈ N be i.i.d. random variables with E Y 2 i = 1 Y i , n ∈ N , and let N be a standard Gaussian random variable, i.e., � x 1 exp ( − u 2 / 2 ) d u , P ( N ≤ x ) = √ x ∈ R . 2 π −∞ Then S n − E S n √ → N n → ∞ , in distribution as Var S n that is, � S n − E S n � √ ≤ x = P ( N ≤ x ) , x ∈ R . n →∞ P lim Var S n June 25, 2015 6/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Classical central limit Theorem Theorem: 1 < ∞ , let S n = � n Let ( Y i ) i ∈ N be i.i.d. random variables with E Y 2 i = 1 Y i , n ∈ N , and let N be a standard Gaussian random variable, i.e., � x 1 exp ( − u 2 / 2 ) d u , P ( N ≤ x ) = √ x ∈ R . 2 π −∞ Then S n − E S n √ → N n → ∞ , in distribution as Var S n that is, � S n − E S n � √ ≤ x = P ( N ≤ x ) , x ∈ R . n →∞ P lim Var S n Does something similar hold for the edge length of the k -nearest neighbour graph or the Poisson-Voronoi tessellation? June 25, 2015 6/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Probability distances For two random variables X 1 and X 2 we define the Kolmogorov distance d K ( X 1 , X 2 ) := sup | P ( X 1 ≤ x ) − P ( X 2 ≤ x ) | x ∈ R and the Wasserstein distance d W ( X 1 , X 2 ) := | E h ( X 1 ) − E h ( X 2 ) | , sup h ∈ Lip ( 1 ) where Lip ( 1 ) is the set of all functions h : R → R with a Lipschitz constant not greater than one. Convergence in d K or d W implies convergence in distribution. June 25, 2015 7/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Berry-Esseen-Bound Theorem: Berry 1941, Esseen 1942 Let ( Y i ) i ∈ N be i.i.d. random variables with E | Y 1 | 3 < ∞ , let S n = � n i = 1 Y i , n ∈ N , and let N be a standard Gaussian random variable. Then there is a constant C > 0 such that E | Y 1 − E Y 1 | 3 � S n − E S n � ≤ C √ , N √ , n ∈ N . d K √ 3 Var S n n Var Y 1 June 25, 2015 8/22 M. Schulte – Central limit theorems for random tessellations and random graphs

Berry-Esseen-Bound Theorem: Berry 1941, Esseen 1942 Let ( Y i ) i ∈ N be i.i.d. random variables with E | Y 1 | 3 < ∞ , let S n = � n i = 1 Y i , n ∈ N , and let N be a standard Gaussian random variable. Then there is a constant C > 0 such that E | Y 1 − E Y 1 | 3 � S n − E S n � ≤ C √ , N √ , n ∈ N . d K √ 3 Var S n n Var Y 1 Aim of this talk: Berry-Esseen bounds for problems from stochastic geometry June 25, 2015 8/22 M. Schulte – Central limit theorems for random tessellations and random graphs

k-Nearest Neighbour Graph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● June 25, 2015 9/22 M. Schulte – Central limit theorems for random tessellations and random graphs

k-Nearest Neighbour Graph η t homogeneous Poisson process of intensity t in a compact convex set H = 1 L ( α ) � 1 { edge between x 1 and x 2 in NNG k ( η t ) }� x 1 − x 2 � α t 2 ( x 1 , x 2 ) ∈ η 2 t , � = Theorem: Last/Peccati/S. 2014+ Let N be a standard Gaussian random variable. Then there are constants C α , α ≥ 0, only depending on k , H and α such that    L ( α ) − E L ( α ) t t  ≤ C α t − 1 / 2 , , N t ≥ 1 . d K � Var L ( α ) t This improves the rates ( ln ( t )) 1 + 3 / 4 t − 1 / 4 by Avram/Bertsimas (1993) and ( ln ( t )) 3 d t − 1 / 2 by Penrose/Yukich (2005). June 25, 2015 10/22 M. Schulte – Central limit theorems for random tessellations and random graphs