## The Annals of Probability

### Multivariate spatial central limit theorems with applications to percolation and spatial graphs

Mathew D. Penrose

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 33, Number 5 (2005), 1945-1991.

Dates
First available in Project Euclid: 22 September 2005

https://projecteuclid.org/euclid.aop/1127395878

Digital Object Identifier
doi:10.1214/009117905000000206

Mathematical Reviews number (MathSciNet)
MR2165584

Zentralblatt MATH identifier
1087.60022

#### Citation

Penrose, Mathew D. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005), no. 5, 1945--1991. doi:10.1214/009117905000000206. https://projecteuclid.org/euclid.aop/1127395878

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