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Normal approximation for statistics of Gibbsian input in geometric probability

Published online by Cambridge University Press:  21 March 2016

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 Wλ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Qλd. 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.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2015 

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