Gaussian Limts for Random Geometric Measures

By Mathew D. Penrose. 2007

79 citations recorded by the source corpus.

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $\nu_n(f)$, and deduce weak convergence of $\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dcb90a3005b6e2c2a86a3c764ac6b56f01d22f1f

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Citations in the corpus

Normal approximation for stabilizing functionals (2017, methodology): “tabilization methods, the authors of [19] proved asymptotic normality for the total edge length of the Euclidean minimal spanning tree, though they did not obtain a rate of convergence. In the papers [6, 24, 27] abstract central limit theorems for stabilizing functionals are derived and applied to s”
Lectures on the Poisson Process (2017, background): “Many Poisson functionals arising in stochastic geometry have a property of stabilisation (local dependence); central limit and normal approximation theorems based on this property have been established in [128, 130, 131].”
Limit theory for point processes in manifolds (2011, methodology): “In keeping with notation from [5, 35], for all such functionals ξ we set”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “al approximation of: (i) statistics (including the total edge count and the total length) based on a k-nearest neighbour graph with a Poisson input (generalising and improving previous estimates from [3, 5, 6, 40, 42, 44]); (ii) the intrinsic volumes of k-faces associated with a Poisson-Voronoi tess”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, background): “ties were established. The surface area was investigated in [19], and the results were extended in [9] to more general functionals and point processes. Integrals over Boolean models are considered in [1, 24], where the volume is included as a special case and also the surface area in the latter one.”