### Normal approximation for statistics of Gibbsian input in geometric probability

#### 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

https://projecteuclid.org/euclid.aap/1449859795

Digital Object Identifier
doi:10.1239/aap/1449859795

Mathematical Reviews number (MathSciNet)
MR3433291

Zentralblatt MATH identifier
1333.60037

#### 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

#### References

• Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.
• Bai, Z.-D., Chao, C.-C., Hwang, H.-K. and Liang, W.-Q. (1998). On the variance of the number of maxima in random vectors and its applications. Ann. Appl. Prob. 8, 886–895.
• Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Electron J. Prob. 6, 41pp.
• Barbour, A. D. and Xia, A. (2001). The number of two dimensional maxima. Adv. Appl. Prob. 33, 727–750.
• Barbour, A. D. and Xia, A. (2006). Normal approximation for random sums. Adv. Appl. Prob. 38, 693–728.
• Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15, 213–253.
• Blaszczyszyn, B., Dhandapani, Y. and Yukich, J. E. (2015). Normal convergence of geometric statistics of clustering point processes. Preprint.
• Calka, P. and Yukich, J. E. (2014). Variance asymptotics for random polytopes in smooth convex bodies. Prob. Theory Relat. Fields 158, 435–463.
• Calka, P. and Yukich, J. E. (2014). Variance asymptotics and scaling limits for Gaussian polytopes. Prob. Theory Relat. Fields 10.1007/s0040-014-0592-6.
• Calka, P., Schreiber, T. and Yukich, J. E. (2013). Brownian limits, local limits and variance asymptotics for convex hulls in the ball. Ann. Prob. 41, 50–108.
• Chiu, S. N. and Lee, H. Y. (2002). A regularity condition and strong limit theorems for linear birth–growth processes. Math. Nachr. 241, 21–27.
• Chiu, S. N. and Quine, M. P. (1997). Central limit theory for the number of seeds in a growth model in $\R^d$ with inhomogeneous Poisson arrivals. Ann. Appl. Prob. 7, 802–814.
• Chiu, S. N. and Quine, M. P. (2001). Central limit theorem for germination-growth models in $\R^d$ with non-Poisson locations. Adv. Appl. Prob. 33, 751–755.
• Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.
• Devroye, L. (1993). Records, the maximal layer, and uniform distributions in monotone sets. Comput. Math. Appl. 25, 19–31.
• Eichelsbacher, P., Raič, M. and Schreiber, T. (2015). Moderate deviations for stabilizing functionals in geometric probability. Ann. Inst. H. Poincaré Prob. Statist. 51, 89–128.
• Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events, Springer, Berlin.
• Fernández, R., Ferrari, P. A. and Garcia, N. L. (2001). Loss network representation of Peierls contours. Ann. Prob. 29, 902–937.
• Ferrari, P. A., Fernández, R. and Garcia, N. L. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 63–88.
• Holst, L., Quine, M. P. and Robinson, J. (1996). A general stochastic model for nucleation and linear growth. Ann. Appl. Prob. 6, 903–921.
• Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, London.
• Last, G. and Penrose, M. D. (2013). Percolation and limit theory for the Poisson lilypond model. Random Structures Algorithms 42, 226–249.
• Last, G., Peccati, G. and Schulte, M. (2014). Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization. Preprint. Available at http://arxiv.org/abs/1401.7568.
• Martin, Ph. A. and Yalcin, T. (1980). The charge fluctuations in classical Coulomb systems. J. Statist. Phys. 22, 435–463.
• Møller, J. (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.
• Møller, J. (2000). Aspects of spatial statistics, stochastic geometry and Markov chain Monte Carlo. Doctoral thesis, Aalborg University.
• Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.
• Penrose, M. D. (2002). Limit theorems for monotonic particle systems and sequential deposition. Stoch. Process. Appl. 98, 175–197.
• Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
• Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Prob. 12, 989–1035.
• Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.
• Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.
• Penrose, M. D. and Yukich, J. E. (2002). Limit theory for random sequential packing and deposition. Ann. Appl. Prob. 12, 272–301.
• Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.
• Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein's Method and Applications (Lecture Notes Ser. Inst. Math. Sci. Nat. Univ. Singapore 5), Singapore University Press, pp. 37–58.
• Penrose, M. D. and Yukich, J. E. (2013). Limit theory for point processes in manifolds. Ann. Appl. Prob. 23, 2161–2211.
• Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.
• Schreiber, T. and Yukich, J. E. (2013). Limit theorems for geometric functionals of Gibbs point processes. Ann. Inst. H. Poincaré Prob. Statist. 49, 1158–1182.
• Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.
• Wade, A. R. (2007). Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. Appl. Prob. 39, 326–342.
• Yukich, J. E. (2015). Surface order scaling in stochastic geometry. Ann. Appl. Prob. 25, 177–210.