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September 2005 Multivariate spatial central limit theorems with applications to percolation and spatial graphs
Mathew D. Penrose
Ann. Probab. 33(5): 1945-1991 (September 2005). DOI: 10.1214/009117905000000206


Suppose X=(Xx,x in Zd) is a family of i.i.d. variables in some measurable space, B0 is a bounded set in Rd, and for t>1, Ht is a measure on tB0 determined by the restriction of X to lattice sites in or adjacent to tB0. We prove convergence to a white noise process for the random measure on B0 given by td/2(Ht(tA)−EHt(tA)) for subsets A of B0, as t becomes large, subject to H satisfying a “stabilization” condition (whereby the effect of changing X at a single site x is local) but with no assumptions on the rate of decay of correlations. We also give a multivariate central limit theorem for the joint distributions of two or more such measures Ht, and adapt the result to measures based on Poisson and binomial point processes. Applications given include a white noise limit for the measure which counts clusters of critical percolation, a functional central limit theorem for the empirical process of the edge lengths of the minimal spanning tree on random points, and central limit theorems for the on-line nearest-neighbor graph.


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Mathew D. Penrose. "Multivariate spatial central limit theorems with applications to percolation and spatial graphs." Ann. Probab. 33 (5) 1945 - 1991, September 2005.


Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1087.60022
MathSciNet: MR2165584
Digital Object Identifier: 10.1214/009117905000000206

Primary: 60D05 , 60F05
Secondary: 05C80 , 60K35

Keywords: central limit theorem , empirical process , Minimal spanning tree , on-line nearest-neighbor graph , percolation , White noise

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • September 2005
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