Advances in Applied Probability

Normal approximation for statistics of Gibbsian input in geometric probability

Aihua Xia and J. E. Yukich

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Abstract

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 QλRd. 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.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 4 (2015), 934-972.

Dates
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1449859795

Digital Object Identifier
doi:10.1239/aap/1449859795

Mathematical Reviews number (MathSciNet)
MR3433291

Zentralblatt MATH identifier
1333.60037

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G55: Point processes

Keywords
Gibbs point process Stein's method random Euclidean graphs maximal points spatial birth-growth model

Citation

Xia, Aihua; Yukich, J. E. Normal approximation for statistics of Gibbsian input in geometric probability. Adv. in Appl. Probab. 47 (2015), no. 4, 934--972. doi:10.1239/aap/1449859795. https://projecteuclid.org/euclid.aap/1449859795


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