Multivariate normal approximation in geometric probability

Mathew D. Penrose, Andrew R. Wade

Research output: Contribution to journalArticle

7 Citations (Scopus)
53 Downloads (Pure)

Abstract

Consider a measure μλ = Σx ξx δx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in Rd are asymptotically independent normals as λ tends to infinity; here we give an O( λ-1/(2d + ε)) bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.
Original languageEnglish
Pages (from-to)293-326
Number of pages23
JournalJournal of Statistical Theory and Practice
Volume2
Issue number2
Publication statusPublished - 2008

Fingerprint

Geometric Probability
Multivariate Approximation
Normal Approximation
Multivariate Normal
Siméon Denis Poisson
Disjoint
Proximity Graphs
Nearest Neighbor Graph
Exponential Stabilization
Poisson Point Process
Moment Conditions
D-space
Central limit theorem
Rate of Convergence
Infinity
Tend
Statistics
Interval
Model

Keywords

  • multivariate normal approximation
  • geometric probability
  • stabilization
  • central limit theorem
  • Stein's method
  • nearest-neighbour graph
  • statistics

Cite this

Penrose, Mathew D. ; Wade, Andrew R. / Multivariate normal approximation in geometric probability. In: Journal of Statistical Theory and Practice. 2008 ; Vol. 2, No. 2. pp. 293-326.
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Multivariate normal approximation in geometric probability. / Penrose, Mathew D.; Wade, Andrew R.

In: Journal of Statistical Theory and Practice, Vol. 2, No. 2, 2008, p. 293-326.

Research output: Contribution to journalArticle

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AU - Penrose, Mathew D.

AU - Wade, Andrew R.

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N2 - Consider a measure μλ = Σx ξx δx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in Rd are asymptotically independent normals as λ tends to infinity; here we give an O( λ-1/(2d + ε)) bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

AB - Consider a measure μλ = Σx ξx δx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in Rd are asymptotically independent normals as λ tends to infinity; here we give an O( λ-1/(2d + ε)) bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

KW - multivariate normal approximation

KW - geometric probability

KW - stabilization

KW - central limit theorem

KW - Stein's method

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