Multivariate normal approximation in geometric probability

Penrose, Mathew D. and Wade, Andrew R. (2008) Multivariate normal approximation in geometric probability. Journal of Statistical Theory and Practice, 2 (2). pp. 293-326. ISSN 1559-8608 (https://doi.org/10.1080/15598608.2008.10411876)

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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.

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