{
  "id": "math/0410021",
  "version": "v1",
  "published": "2004-10-01T14:58:27.000Z",
  "updated": "2004-10-01T14:58:27.000Z",
  "title": "Multivariate spatial central limit theorems with applications to percolation and spatial graphs",
  "authors": [
    "Mathew D Penrose"
  ],
  "comment": "46 pages. 1 diagram",
  "categories": [
    "math.PR"
  ],
  "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 $tB_0$ determined by the restriction of $X$ to lattice sites in or adjacent to $tB_0$. 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 neighbour graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-01T14:58:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60D05",
      "05C80",
      "60K35"
    ],
    "keywords": [
      "multivariate spatial central limit theorems",
      "spatial graphs",
      "percolation",
      "applications",
      "multivariate central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10021P"
    }
  }
}