Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T04:46:59.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal approximation for statistics of Gibbsian input in geometric probability

Part of: Geometric probability and stochastic geometry Stochastic processes Limit theorems

Published online by Cambridge University Press:  21 March 2016

Aihua Xia  and
J. E. Yukich
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: [email protected]
∗∗ Postal address: Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

Keywords

Gibbs point processStein's methodrandom Euclidean graphsmaximal pointsspatial birth-growth model

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 934 - 972
Copyright
Copyright © Applied Probability Trust 2015 

References

Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Barbour, A. D. and Xia, A. (2001). The number of two dimensional maxima. Adv. Appl. Prob. 33, 727–750.Google Scholar
Barbour, A. D. and Xia, A. (2006). Normal approximation for random sums. Adv. Appl. Prob. 38, 693–728.Google Scholar
Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15, 213–253.CrossRefGoogle Scholar
Blaszczyszyn, B., Dhandapani, Y. and Yukich, J. E. (2015). Normal convergence of geometric statistics of clustering point processes. Preprint.Google Scholar
Calka, P. and Yukich, J. E. (2014). Variance asymptotics for random polytopes in smooth convex bodies. Prob. Theory Relat. Fields 158, 435–463.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Devroye, L. (1993). Records, the maximal layer, and uniform distributions in monotone sets. Comput. Math. Appl. 25, 19–31.Google Scholar
Eichelsbacher, P., Raiŭ, M. and Schreiber, T. (2015). Moderate deviations for stabilizing functionals in geometric probability. Ann. Inst. H. Poincaré Prob. Statist. 51, 89–128.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events, Springer, Berlin.Google Scholar
Fernández, R., Ferrari, P. A. and Garcia, N. L. (2001). Loss network representation of Peierls contours. Ann. Prob. 29, 902–937.Google Scholar
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.Google Scholar
Holst, L., Quine, M. P. and Robinson, J. (1996). A general stochastic model for nucleation and linear growth. Ann. Appl. Prob. 6, 903-921.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Academic Press, London.CrossRefGoogle Scholar
Last, G. and Penrose, M. D. (2013). Percolation and limit theory for the Poisson lilypond model. Random Structures Algorithms 42, 226–249.CrossRefGoogle Scholar
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.Google Scholar
Martin, Ph. A. and Yalcin, T. (1980). The charge fluctuations in classical Coulomb systems. J. Statist. Phys. 22, 435–463.Google Scholar
Møller, J. (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.Google Scholar
Møller, J., (2000). Aspects of spatial statistics, stochastic geometry and Markov chain Monte Carlo. Doctoral thesis, Aalborg University.Google Scholar
M⊘ller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Penrose, M. D. (2002). Limit theorems for monotonic particle systems and sequential deposition. Stoch. Process. Appl. 98, 175–197.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Prob. 12, 989–1035.Google Scholar
Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124-1150.Google Scholar
Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.Google Scholar
Penrose, M. D. and Yukich, J. E. (2002). Limit theory for random sequential packing and deposition. Ann. Appl. Prob. 12, 272–301.Google Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar
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.Google Scholar
Penrose, M. D. and Yukich, J. E. (2013). Limit theory for point processes in manifolds. Ann. Appl. Prob. 23, 2161–2211.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
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.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Wade, A. R. (2007). Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. Appl. Prob. 39, 326–342.Google Scholar
Yukich, J. E. (2015). Surface order scaling in stochastic geometry. Ann. Appl. Prob. 25, 177–210.Google Scholar