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

• 成果类型: Article
• 署名作者: Penrose, MD
• 署名单位: University of Bath
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000206
• 发表日期: 2005
• 页码: 1945-1991
• 关键词: minimal spanning-trees; asymptotics; LAWS

摘要

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 B0 given by t(-d/2)(H-t(tA) - EHt (tA)) 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.