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

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 (tA) - EH t (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.

[1]  J. E. Yukich,et al.  Gaussian Limits for Random Measures in Geometric Probability , 2005, math/0503474.

[2]  Sungchul Lee,et al.  The central limit theorem for Euclidean minimal spanning trees II , 1999, Advances in Applied Probability.

[3]  Ronald Meester,et al.  Continuum percolation: Preface , 1996 .

[4]  High density asymptotics of the Poisson random connection model , 2003 .

[5]  J. Yukich,et al.  Central limit theorems for some graphs in computational geometry , 2001 .

[6]  Sungchul Lee,et al.  The central limit theorem for Euclidean minimal spanning trees I I , 1997 .

[7]  Joseph E. Yukich.  Asymptotics for weighted minimal spanning trees on random points , 2000 .

[8]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[9]  Ilya S. Molchanov,et al.  Central limit theorem for a class of random measures associated with germ-grain models , 1996, Advances in Applied Probability.

[10]  J. Yukich,et al.  Weak laws of large numbers in geometric probability , 2003 .

[11]  Yu Zhang.  A Martingale Approach in the Study of Percolation Clusters on the Zd Lattice , 2001 .

[12]  Christos H. Papadimitriou,et al.  Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet , 2002, ICALP.

[13]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[14]  D. McLeish.  Dependent Central Limit Theorems and Invariance Principles , 1974 .

[15]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[16]  Harry Kesten,et al.  The central limit theorem for weighted minimal spanning trees on random points , 1996 .

[17]  Mathew D. Penrose,et al.  Normal Approximation in Geometric Probability , 2004 .

[18]  R. Durrett.  Probability: Theory and Examples , 1993 .

[19]  J. Yukich.  Probability theory of classical Euclidean optimization problems , 1998 .

[20]  M. Penrose,et al.  Large deviations for discrete and continuous percolation , 1996, Advances in Applied Probability.

[21]  Mathew D. Penrose,et al.  On the total length of the random minimal directed spanning tree , 2004, Advances in Applied Probability.

[22]  J. Michael Steele,et al.  Asymptotics for Euclidean minimal spanning trees on random points , 1992 .

[23]  Geoffrey Grimmett,et al.  Central Limit Theorems for Associated Random Variables and the Percolation Model , 1984 .