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Normal approximation for statistics of Gibbsian input in geometric probability

Part of:Geometric probability and stochastic geometryLimit theoremsStochastic processes

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: aihuaxia@unimelb.edu.au
∗∗ Postal address: Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. Email address: joseph.yukich@lehigh.edu
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Abstract

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This paper concerns the asymptotic behavior of a random variable W λ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Q λ ↑ ℝ d . We establish conditions ensuring that W λ 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 W λ 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 geometry60G55: Point processes

Information

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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