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Normal approximation for statistics of Gibbsian input in geometric probability

Part of: Limit theorems Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  21 March 2016

Aihua Xia  and
J. E. Yukich
Aihua Xia*
Affiliation:
The University of Melbourne
J. E. Yukich*
Affiliation:
Lehigh University
*
∗ Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. Email address: [email protected]
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Abstract

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This paper concerns the asymptotic behavior of a random variable Wλ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Qλ ↑ ℝd. We establish conditions ensuring that Wλ has volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation for Wλ as λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish a similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

Keywords

Gibbs point processStein's methodrandom Euclidean graphsmaximal pointsspatial birth-growth model

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 934 - 972
Copyright
Copyright © Applied Probability Trust 2015 

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