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

Abstract

Suppose X = (X x, x in Z d) is a family of i.i.d. variables in some measurable space, B 0 is a bounded set in R d, and for t > 1, H t is a measure on t B 0 determined by the restriction of X to lattice sites in or adjacent to t B 0. We prove convergence to a white noise process for the random measure on B 0 given by t -d/2(H t(t A) - E H t(t A)) for subsets A of B 0, 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 H t, 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. © Institute of Mathematical Statistics, 2005.