Multivariate normal approximation in geometric probability

Mathew D. Penrose, Andrew R. Wade

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9 Citations (Scopus)
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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
Issue number2
Publication statusPublished - 2008


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


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