Source: Ann. Appl. Probab.
Volume 13, Number 1
Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers functionals of marked point processes, including statistics of Boolean models.
 ALDOUS, D. and STEELE, J. M. (1992). Asy mptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
 Alexander, K. S. (1995). Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phy s. 168 39-55.
 BEZUIDENHOUT, C., GRIMMETT, G. and LÖFFLER, A. (1998). Percolation and minimal spanning trees. J. Statist. Phy s. 92 1-34.
 BORG, I. and GROENEN, P. (1997). Modern Multidimensional Scaling: Theory and Applications. Springer, New York.
 BRITO, M., QUIROZ, A. and YUKICH, J. E. (2002). Graph theoretic procedures for dimension identification. J. Multivariate Anal. 81 67-84.
 DEVROy E, L. (1988). The expected size of some graphs in computational geometry. Comput. Math. Appl. 15 53-64.
 EPPSTEIN, D., PATERSON, M. S. and YAO, F. F. (1997). On nearest-neighbor graphs. Discrete Comput. Geom. 17 263-282.
 FÜREDI, Z. (1997). The expected size of a random sphere of influence graph. In Intuitive Geometry (I. Bárány and K. Böröczky, eds.) 319-326. János Boly ai Math. Society, Budapest, Hungary.
 GRIMMETT, G. (1989, 1999). Percolation. Springer, New York.
 HALL, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
Mathematical Reviews (MathSciNet): MR973404
 HAy EN, A. and QUINE, M. P. (2000). The proportion of triangles in a Poisson-Voronoi tessellation of the plane. Adv. in Appl. Probab. 32 67-74.
 HENZE, N. (1987). On the fraction of random points with specified nearest neighbor interrelations and degree of attraction. Adv. in Appl. Probab. 19 873-895.
Mathematical Reviews (MathSciNet): MR914597
 HITCZENKO, P., JANSON, S. and YUKICH, J. E. (1999). On the variance of the random sphere of influence graph. Random Structures Algorithms 14 139-152.
 JAROMCZy K, J. W. and TOUSSAINT, G. T. (1992). Relative neighborhood graphs and their relatives. Proc. IEEE 80 1502-1517.
 JIMENEZ, R. and YUKICH, J. E. (2002). Strong laws for Euclidean graphs with general edge weights. Statist. Probab. Lett. 56 251-259.
 KESTEN, H. and LEE, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
 MATHERON, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
 MCGIVNEY, K. (1997). Probabilistic limit theorems for combinatorial optimization problems. Ph.D. dissertation, Lehigh Univ.
 MCGIVNEY, K. and YUKICH, J. E. (1999). Asy mptotics for Voronoi tessellations on random samples. Stochastic Process. Appl. 83 273-288.
 MOLCHANOV, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, New York.
 PENROSE, M. D. (1996). The random minimal spanning tree in high dimensions. Ann. Probab. 24 1903-1925.
 PENROSE, M. D. (2001). Random parking, sequential adsorption, and the jamming limit. Comm. Math. Phy s. 218 153-176.
 PENROSE, M. D. and YUKICH, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005-1041.
 PENROSE, M. D. and YUKICH, J. E. (2001). Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 272-301.
 RUDIN, W. (1987). Real and Complex Analy sis, 3rd ed. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet): MR924157
 SMITH, W. D. (1989). Studies in computational geometry motivated by mesh generation. Ph.D. dissertation, Princeton Univ.
 STEELE, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia.
 STEELE, J. M., SHEPP, L. and EDDY, W. (1987). On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab. 24 809-826.
 YUKICH, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.
 YUKICH, J. E. (1999). Asy mptotics for weighted minimal spanning trees on random points. Stochastic Process. Appl. 85 123-138.
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