Gaussian Limits for Random Measures in Geometric Probability

By yuliy baryshnikov, Joseph E. Yukich. 2005

105 citations recorded by the source corpus.

We establish Gaussian limits for measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random graphs (nearest neighbor, Voronoi, and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth growth models), and statistics of germ-grain models.

Subfields: Applied Mathematics, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2698f9596b1d5d727386741a7ea7d3c634b1a409

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Citations in the corpus

Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, background): “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.”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, background): “The CLTs presented for instance in [1, 29] cover well this case.”
Laws of large numbers in stochastic geometry with statistical applications (2007, background): “Many other fields of application have been discussed elsewhere [2,16,18] and we do not attempt to review all of these, but our results often enable us to generalize results in those papers as indicated above.”
Normal approximation for stabilizing functionals (2017, background): “For example, in the setting X = R, we expect that all of the statistics Hs and H ′ n described in [6, 25, 28, 29, 30] consist of sums of scores ξs and ξn satisfying the conditions of Theorem 2.”
Gaussian Limts for Random Geometric Measures (2007, methodology): “Recently, attention has turned to the asymptotic behaviour of the measure itself (rather than only its total measure), notably in (3; 11; 18; 19; 24; 25).”
Normal Approximation in Geometric Probability (2004, methodology): “1 Introduction In the study of limit theorems for functionals on Poisson or binomial spatial point processes, the notion of stabilization has recently proved to be a useful unifying concept [4, 11, 13].”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “Central limit theorems for the total edge length of the k-nearest neighbour graph were studied in [3, 5, 6, 40, 42, 44].”
Limit theory for point processes in manifolds (2011, methodology): “In keeping with notation from [5, 35], for all such functionals ξ we set”