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The characteristic independence property of Poisson point processes gives an intuitive way to explain why a sequence of point processes becoming less and less repulsive can converge to a Poisson point process. The aim of this paper is to show this convergence for sequences built by superposing, thinning or rescaling determinantal processes. We use Papangelou intensities and Stein’s method to prove this result with a topology based on total variation distance.

Asymptotics of superposition of point processes

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        <identifier identifierType="DOI">10.23723/11784/14260</identifier><creators><creator><creatorName>Laurent Decreusefond</creatorName></creator><creator><creatorName>Aurélien Vasseur</creatorName></creator></creators><titles>
            <title>Asymptotics of superposition of point processes</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Poisson point process</subject><subject>Stochastic geometry</subject><subject>Ginibre point process</subject><subject>β-Ginibre point process</subject><subject>Steinâs method</subject></subjects><dates>
	    <date dateType="Created">Sat 7 Nov 2015</date>
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The characteristic independence property of Poisson point processes gives an intuitive way to explain why a sequence of point processes becoming less and less repulsive can converge to a Poisson point process. The aim of this paper is to show this convergence for sequences built by superposing, thinning or rescaling determinantal processes. We use Papangelou intensities and Steinâs method to prove this result with a topology based on total variation distance.

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I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications 2nd conference on Geometric Science of Information Aurélien VASSEUR Asymptotics of some Point Processes Transformations Ecole Polytechnique, Paris-Saclay, October 28, 2015 1/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Mobile network in Paris - Motivation −2000 0 2000 4000 100020003000 −2000 0 2000 4000 100020003000 Figure: On the left, positions of all BS in Paris. On the right, locations of BS for one frequency band. 2/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Table of Contents I-Generalities on point processes Correlation function, Papangelou intensity and repulsiveness Determinantal point processes II-Kantorovich-Rubinstein distance Convergence dened by dKR dKR(PPP, Φ) ≤ "nice" upper bound III-Applications to transformations of point processes Superposition Thinning Rescaling 3/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Framework Y a locally compact metric space µ a diuse and locally nite measure of reference on Y NY the space of congurations on Y NY the space of nite congurations on Y 4/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Correlation function - Papangelou intensity Correlation function ρ of a point process Φ: E[ α∈NY α⊂Φ f (α)] = +∞ k=0 1 k! ˆ Yk f · ρ({x1, . . . , xk})µ(dx1) . . . µ(dxk) ρ(α) ≈ probability of nding a point in at least each point of α Papangelou intensity c of a point process Φ: E[ x∈Φ f (x, Φ \ {x})] = ˆ Y E[c(x, Φ)f (x, Φ)]µ(dx) c(x, ξ) ≈ conditionnal probability of nding a point in x given ξ 5/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Point process Properties Intensity measure: A ∈ FY → ´ A ρ({x})µ(dx) ρ({x}) = E[c(x, Φ)] If Φ is nite, then: IP(|Φ| = 1) = ˆ Y c(x, ∅)µ(dx) IP(|Φ| = 0). 6/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Poisson point process Properties Φ PPP with intensity M(dy) = m(y)dy Correlation function: ρ(α) = x∈α m(x) Papangelou intensity: c(x, ξ) = m(x) 7/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Repulsive point process Denition Point process repulsive if φ ⊂ ξ =⇒ c(x, ξ) ≤ c(x, φ) Point process weakly repulsive if c(x, ξ) ≤ c(x, ∅) 8/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Determinantal point process Denition Determinantal point process DPP(K, µ): ρ({x1, · · · , xk}) = det(K(xi , xj ), 1 ≤ i, j ≤ k) Proposition Papangelou intensity of DPP(K, µ): c(x0, {x1, · · · , xk}) = det(J(xi , xj ), 0 ≤ i, j ≤ k) det(J(xi , xj ), 1 ≤ i, j ≤ k) where J = (I − K)−1K. 9/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process Ginibre point process Denition Ginibre point process on B(0, R): K(x, y) = 1 π e−1 2 (|x|2 +|y|2 ) exy 1{x∈B(0,R)}1{y∈B(0,R)} β-Ginibre point process on B(0, R): Kβ(x, y) = 1 π e − 1 2β (|x|2 +|y|2 ) e 1 β xy 1{x∈B(0,R)} 1{y∈B(0,R)} 10/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Framework Determinantal point process β-Ginibre point processes 11/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Kantorovich-Rubinstein distance Total variation distance: dTV(ν1, ν2) := sup A∈FY ν1(A),ν2(A)<∞ |ν1(A) − ν2(A)| F : NY → IR is 1-Lipschitz (F ∈ Lip1) if |F(φ1) − F(φ2)| ≤ dTV (φ1, φ2) for all φ1, φ2 ∈ NY Kantorovich-Rubinstein distance: dKR(IP1, IP2) = sup F∈Lip1 ˆ NY F(φ) IP1(dφ) − ˆ NY F(φ) IP2(dφ) Convergence in K.-R. distance =⇒ strictly Convergence in law 12/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Upper bound theorem Theorem (L. Decreusefond, AV) Φ a nite point process on Y ζM a PPP with nite control measure M(dy) = m(y)µ(dy). Then, we have: dKR(IPΦ, IPζM ) ≤ ˆ Y ˆ NY |m(y) − c(y, φ)|IPΦ(dφ)µ(dy). 13/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Superposition of weakly repulsive point processes Φn,1, . . . , Φn,n: n independent, nite and weakly repulsive point processes on Y Φn := n i=1 Φn,i Rn := ´ Y | n i=1 ρn,i (x) − m(x)|µ(dx) ζM a PPP with control measure M(dx) = m(x)µ(dx) 14/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Superposition of weakly repulsive point processes Proposition (LD, AV) Φn = n i=1 Φn,i ζM a PPP with control measure M(dx) = m(x)µ(dx) dKR(IPΦn , IPζM ) ≤ Rn + max 1≤i≤n ˆ Y ρn,i (x)µ(dx) 15/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Consequence Corollary (LD, AV) f pdf on [0; 1] such that f (0+) := limx→0+ f (x) ∈ IR Λ compact subset of IR+ X1, . . . , Xn i.i.d. with pdf fn = 1 n f (1 n ·) Φn = {X1, . . . , Xn} ∩ Λ dKR(Φn, ζ) ≤ ˆ Λ f 1 n x − f (0+) dx + 1 n ˆ Λ f 1 n x dx where ζ is the PPP(f (0+)) reduced to Λ. 16/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning β-Ginibre point processes Proposition (LD, AV) Φn the βn-Ginibre process reduced to a compact set Λ ζ the PPP with intensity 1/π on Λ dKR(IPΦn , IPζ) ≤ Cβn 17/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Kallenberg's theorem Theorem (O. Kallenberg) Φn a nite point process on Y pn : Y → [0; 1) uniformly −−−−−→ 0 Φn the pn-thinning of Φn γM a Cox process (pnΦn) law −−→ M ⇐⇒ (Φn) law −−→ γM 18/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Polish distance (fn) a sequence in the space of real continuous functions with compact support generating FY d∗(ν1, ν2) = n≥1 1 2n Ψ(|ν1(fn) − ν2(fn)|) with Ψ(x) = x 1 + x d∗ KR the Kantorovich-Rubinstein distance associated to the distance d∗ 19/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning Thinned point processes Proposition (LD, AV) Φn a nite point process on Y pn : Y → [0; 1) Φn the pn-thinning of Φn γM a Cox process Then, we have: d∗ KR(IPΦn , IPγM ) ≤ 2E[ x∈Φn p2 n(x)] + d∗ KR(IPM, IPpnΦn ). 20/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Application to superposition Application to β-Ginibre point processes Application to thinning References L.Decreusefond, and A.Vasseur, Asymptotics of superposition of point processes, 2015. H.O. Georgii, and H.J. Yoo, Conditional intensity and gibbsianness of determinantal point processes, J. Statist. Phys. (118), January 2004. J.S. Gomez, A. Vasseur, A. Vergne, L. Decreusefond, P. Martins, and Wei Chen, A Case Study on Regularity in Cellular Network Deployment, IEEE Wireless Communications Letters, 2015. A.F. Karr, Point Processes and their Statistical Inference, Ann. Probab. 15 (1987), no. 3, 12261227. 21/22 Aurélien VASSEUR Télécom ParisTech I-Generalities on point processes II-Kantorovich-Rubinstein distance III-Applications Thank you ... ... for your attention. Questions? 22/22 Aurélien VASSEUR Télécom ParisTech