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 t−d/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.
"Multivariate spatial central limit theorems with applications to percolation and spatial graphs." Ann. Probab. 33 (5) 1945 - 1991, September 2005. https://doi.org/10.1214/009117905000000206