Rate of Poisson approximation for nearest neighbor counts in large-dimensional Poisson point processes
Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be the fraction of Π-points which are the nearest Π-neighbor of precisely k Π′-points. It is known that as d→∞, the qk,d converge to the Poisson...
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| Published in | Statistics & probability letters Vol. 92; pp. 143 - 147 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.09.2014
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| Subjects | |
| Online Access | Get full text |
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| Abstract | Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be the fraction of Π-points which are the nearest Π-neighbor of precisely k Π′-points. It is known that as d→∞, the qk,d converge to the Poisson probabilities e−λ′/λ(λ′/λ)k/k!, k=0,1,…. We derive the (sharp) rate of convergence d−1/2(4/33)d, which is related to the asymptotic behavior of the variance of the volume of the typical cell of the Poisson–Voronoi tessellation generated by Π. An extension to the case involving more than two independent Poisson point processes is also considered. |
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| AbstractList | Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be the fraction of Π-points which are the nearest Π-neighbor of precisely k Π′-points. It is known that as d→∞, the qk,d converge to the Poisson probabilities e−λ′/λ(λ′/λ)k/k!, k=0,1,…. We derive the (sharp) rate of convergence d−1/2(4/33)d, which is related to the asymptotic behavior of the variance of the volume of the typical cell of the Poisson–Voronoi tessellation generated by Π. An extension to the case involving more than two independent Poisson point processes is also considered. |
| Author | Yao, Yi-Ching |
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| Cites_doi | 10.1239/aap/1275055233 10.2307/1427321 10.1239/aap/1231340158 10.2307/1427088 10.1214/aoap/1034968144 |
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| Copyright | 2014 Elsevier B.V. |
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| Keywords | secondary Typical cell Voronoi tessellation Typical point Total variation distance primary |
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| References | Barbour, Holst, Janson (br000010) 1992 Daley, Vere-Jones (br000015) 2007 Yao (br000035) 2010; 42 Newman, Rinott, Tversky (br000025) 1983; 15 Alishahi, Sharifitabar (br000005) 2008; 40 Newman, Rinott (br000020) 1985; 17 Yao, Simons (br000040) 1996; 6 Okabe, Boots, Sugihara, Chiu (br000030) 2000 Yao (10.1016/j.spl.2014.05.014_br000035) 2010; 42 Yao (10.1016/j.spl.2014.05.014_br000040) 1996; 6 Barbour (10.1016/j.spl.2014.05.014_br000010) 1992 Okabe (10.1016/j.spl.2014.05.014_br000030) 2000 Daley (10.1016/j.spl.2014.05.014_br000015) 2007 Newman (10.1016/j.spl.2014.05.014_br000020) 1985; 17 Alishahi (10.1016/j.spl.2014.05.014_br000005) 2008; 40 Newman (10.1016/j.spl.2014.05.014_br000025) 1983; 15 |
| References_xml | – volume: 42 start-page: 359 year: 2010 end-page: 370 ident: br000035 article-title: On variances of partial volumes of the typical cell of a Poisson–Voronoi tessellation and large-dimensional volume degeneracy publication-title: Adv. Appl. Probab. contributor: fullname: Yao – volume: 15 start-page: 726 year: 1983 end-page: 751 ident: br000025 article-title: Nearest neighbors and Voronoi regions in certain point processes publication-title: Adv. Appl. Probab. contributor: fullname: Tversky – volume: 17 start-page: 794 year: 1985 end-page: 809 ident: br000020 article-title: Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions publication-title: Adv. Appl. Probab. contributor: fullname: Rinott – volume: 40 start-page: 919 year: 2008 end-page: 938 ident: br000005 article-title: Volume degeneracy of the typical cell and the chord length distribution for Poisson–Voronoi tessellations in high dimensions publication-title: Adv. Appl. Probab. contributor: fullname: Sharifitabar – year: 1992 ident: br000010 article-title: Poisson Approximation contributor: fullname: Janson – year: 2007 ident: br000015 article-title: An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure contributor: fullname: Vere-Jones – volume: 6 start-page: 561 year: 1996 end-page: 571 ident: br000040 article-title: A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes publication-title: Ann. Appl. Probab. contributor: fullname: Simons – year: 2000 ident: br000030 article-title: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams contributor: fullname: Chiu – volume: 42 start-page: 359 year: 2010 ident: 10.1016/j.spl.2014.05.014_br000035 article-title: On variances of partial volumes of the typical cell of a Poisson–Voronoi tessellation and large-dimensional volume degeneracy publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1275055233 contributor: fullname: Yao – volume: 15 start-page: 726 year: 1983 ident: 10.1016/j.spl.2014.05.014_br000025 article-title: Nearest neighbors and Voronoi regions in certain point processes publication-title: Adv. Appl. Probab. doi: 10.2307/1427321 contributor: fullname: Newman – year: 1992 ident: 10.1016/j.spl.2014.05.014_br000010 contributor: fullname: Barbour – year: 2000 ident: 10.1016/j.spl.2014.05.014_br000030 contributor: fullname: Okabe – volume: 40 start-page: 919 year: 2008 ident: 10.1016/j.spl.2014.05.014_br000005 article-title: Volume degeneracy of the typical cell and the chord length distribution for Poisson–Voronoi tessellations in high dimensions publication-title: Adv. Appl. Probab. doi: 10.1239/aap/1231340158 contributor: fullname: Alishahi – year: 2007 ident: 10.1016/j.spl.2014.05.014_br000015 contributor: fullname: Daley – volume: 17 start-page: 794 year: 1985 ident: 10.1016/j.spl.2014.05.014_br000020 article-title: Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions publication-title: Adv. Appl. Probab. doi: 10.2307/1427088 contributor: fullname: Newman – volume: 6 start-page: 561 year: 1996 ident: 10.1016/j.spl.2014.05.014_br000040 article-title: A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes publication-title: Ann. Appl. Probab. doi: 10.1214/aoap/1034968144 contributor: fullname: Yao |
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| Snippet | Consider two independent homogeneous Poisson point processes Π of intensity λ and Π′ of intensity λ′ in d-dimensional Euclidean space. Let qk,d, k=0,1,…, be... |
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| SubjectTerms | Total variation distance Typical cell Typical point Voronoi tessellation |
| Title | Rate of Poisson approximation for nearest neighbor counts in large-dimensional Poisson point processes |
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