Advances in Applied Probability

Normal approximation for statistics of Gibbsian input in geometric probability

Aihua Xia and J. E. Yukich

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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 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

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

First available in Project Euclid: 11 December 2015

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Zentralblatt MATH identifier

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

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


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.

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