Multivariate Normal Approximation in Geometric Probability
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Consider a measure μλ = Σxξxδx 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 → ℞ here we give an O(λ−1/(2d+ε)) bound on the rate of convergence, and also a new criterion for the limiting normals to be non-degenerate. 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.
Key-wordsMultivariate normal approximation geometric probability stabilization central limit theorem Stein’s method nearest-neighbour graph
AMS Subject Classification60D05 60F05 60G57
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