## Electronic Journal of Probability

### Gaussian Limts for Random Geometric Measures

Mathew Penrose

#### Abstract

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $\nu_n(f)$, and deduce weak convergence of $\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 35, 989-1035.

Dates
Accepted: 2 August 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818506

Digital Object Identifier
doi:10.1214/EJP.v12-429

Mathematical Reviews number (MathSciNet)
MR2336596

Keywords
Random measures

Rights

#### Citation

Penrose, Mathew. Gaussian Limts for Random Geometric Measures. Electron. J. Probab. 12 (2007), paper no. 35, 989--1035. doi:10.1214/EJP.v12-429. https://projecteuclid.org/euclid.ejp/1464818506

#### References

• Avram, Florin; Bertsimas, Dimitris. On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 (1993), no. 4, 1033–1046.
• Yu. Baryshnikov, M. D. Penrose and J. E. Yukich (2005). Gaussian limits for generalized spacings. In preparation.
• Baryshnikov, Yu.; Yukich, J. E. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005), no. 1A, 213–253.
• Bickel, Peter J.; Breiman, Leo. Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 (1983), no. 1, 185–214.
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• Evans, Dafydd; Jones, Antonia J. A proof of the gamma test. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2027, 2759–2799.
• J. W. Evans (1993). Random and cooperative adsorption, Reviews of Modern Physics, 65, 1281-1329.
• Götze, F.; Heinrich, L.; Hipp, C. $m$-dependent random fields with analytic cumulant generating function. Scand. J. Statist. 22 (1995), no. 2, 183–195.
• Hall, Peter. Introduction to the theory of coverage processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1988. xx+408 pp. ISBN: 0-471-85702-5
• Heinrich, Lothar. Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 40 (1993), no. 2, 67–94.
• Heinrich, Lothar; Molchanov, Ilya S. Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 (1999), no. 2, 283–314.
• Kingman, J. F. C. Poisson processes. Oxford Studies in Probability, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. viii+104 pp. ISBN: 0-19-853693-3
• Mase, Shigeru. Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Probab. 19 (1982), no. 1, 111–126.
• McGivney, K.; Yukich, J. E. Asymptotics for Voronoi tessellations on random samples. Stochastic Process. Appl. 83 (1999), no. 2, 273–288.
• Meester, Ronald; Roy, Rahul. Continuum percolation. Cambridge Tracts in Mathematics, 119. Cambridge University Press, Cambridge, 1996. x+238 pp. ISBN: 0-521-47504-X
• Molchanov, Ilya; Stoyan, Dietrich. Asymptotic properties of estimators for parameters of the Boolean model. Adv. in Appl. Probab. 26 (1994), no. 2, 301–323.
• Penrose, Mathew. Random geometric graphs. Oxford Studies in Probability, 5. Oxford University Press, Oxford, 2003. xiv+330 pp. ISBN: 0-19-850626-0
• Penrose, Mathew D. Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005), no. 5, 1945–1991.
• Penrose, Mathew D. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007), no. 4, 1124–1150.
• Penrose, Mathew D.; Wade, Andrew R. On the total length of the random minimal directed spanning tree. Adv. in Appl. Probab. 38 (2006), no. 2, 336–372.
• Penrose, Mathew D.; Yukich, J. E. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001), no. 4, 1005–1041.
• Penrose, Mathew D.; Yukich, J. E. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002), no. 1, 272–301.
• Penrose, Mathew D.; Yukich, J. E. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003), no. 1, 277–303.
• Penrose, Mathew D.; Yukich, J. E. Normal approximation in geometric probability. Stein's method and applications, 37–58, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005.
• Schreiber, T.; Yukich, J. E. Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs. Stochastic Process. Appl. 115 (2005), no. 8, 1332–1356.
• Stoyan, D.; Kendall, W. S.; Mecke, J. Stochastic geometry and its applications. With a foreword by D. G. Kendall. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. 345 pp. ISBN: 0-471-90519-4
• Wade, Andrew R. Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. in Appl. Probab. 39 (2007), no. 2, 326–342.