Journal article
Normal approximation for statistics of Gibbsian input in geometric probability
A Xia, JE Yukich
Advances in Applied Probability | APPLIED PROBABILITY TRUST | Published : 2015
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 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..
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Funding Acknowledgements
The research of A. Xia was supported in part by the Australian Research Council Discovery Project (grant nos. DP130101123 and DP150101459). The research of J. E. Yukich was supported in part by the National Science Foundation (grants no. DMS-1106619 and DMS-1406410). We thank Yogeshwaran Dhandapani for showing us Lemma 4 6, which essentially first appeared in [7]. We also thank the anonymous referees for comments leading to an improved exposition. J. E. Yukich gratefully acknowledges generous and kind support from the Department of Mathematics and Statistics at the University of Melbourne, where this work was initiated.