[en] We consider the Gaussian approximation for functionals of a Poisson process that are expressible as sums of region-stabilizing (determined by the points of the process within some specified regions) score functions and provide a bound on the rate of convergence in the Wasserstein and the Kolmogorov distances. While such results have previously been shown in Lachièze-Rey, Schulte and Yukich (2019), we extend the applicability by relaxing some conditions assumed there and provide further insight into the results. This is achieved by working with stabilization regions that may differ from balls of random radii commonly used in the literature concerning stabilizing functionals. We also allow for non-diffuse intensity measures and unbounded scores, which are useful in some applications. As our main application, we consider the Gaussian approximation of number of minimal points in a homogeneous Poisson process in $[0,1]^d$ with $d \geq 2$, and provide a presumably optimal rate of convergence.
Disciplines :
Mathématiques
Auteur, co-auteur :
BHATTACHARJEE, Chinmoy ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Molchanov, Ilya; University of Bern > IMSV
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Gaussian approximation for sums of region-stabilizing scores
Date de publication/diffusion :
24 juillet 2022
Titre du périodique :
Electronic Journal of Probability
eISSN :
1083-6489
Maison d'édition :
Institute of Mathematical Statistics, Beachwood, Etats-Unis - Ohio
Bai, Z. D., Devroye, L., Hwang, H. K. and Tsai, T. H.: Maxima in hypercubes. Random Struct. Algorithms, 27, (2005), 290–309. MR2162600
Barbour, A. D. and Xia, A.: Normal approximation for random sums. Adv. in Appl. Probab., 38, (2006), 693–728. MR2256874
Baryshnikov, Y.: Supporting-points processes and some of their applications. Probab. Theory Related Fields, 117, (2000), 163–182. MR1771659
Bhattacharjee, C.: Gaussian approximation in random minimal directed spanning trees. Random Struct. Algorithms, 61, (2022), 462–492.
Bhattacharjee, C., Molchanov, I. and Turin, R.: Central limit theorem for birth-growth model with Poisson arrivals and random growth speed. arXiv preprint arXiv:2107.06792, (2021).
Gunnar, E.: A remainder term estimate for the normal approximation in classical occupancy. Ann. Probab., 9, (1981), 684–692. MR0624696
Fill, J. A. and Naiman, D. Q.: The Pareto record frontier. Electron. J. Probab., 25, (2020). MR4136472
Iyer, S. K. and Thacker, D.: Nonuniform random geometric graphs with location-dependent radii. Ann. Appl. Probab., 22, (2012), 2048–2066. MR3025688
Lachièze-Rey, R., Schulte, M. and Yukich, J. E.: Normal approximation for stabilizing function-als. Ann. Appl. Probab., 29, (2019), 931–993. MR3910021
Last, G., Peccati, G. and Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields, 165, (2016), 667–723. MR3520016
Last, G. and Penrose, M.: Lectures on the Poisson Process. Cambridge Univ. Press, Cambridge, 2018. MR3791470
Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford, 2003. MR1986198
Penrose, M. D. and Yukich, J. E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab., 11, (2001), 1005–1041. MR1878288
Penrose, M. D. and Yukich, J. E.: Weak laws of large numbers in geometric probability. Ann. Appl. Probab., 13, (2003), 277–303. MR1952000
Penrose, M. D. and Yukich, J. E.: Normal approximation in geometric probability. In Stein’s Method and Applications, volume 5 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 37–58. Singapore Univ. Press, Singapore, 2005. MR2201885
Schreiber, T.: Limit theorems in stochastic geometry. In W. S. Kendall and I. Molchanov, editors, New Perspectives in Stochastic Geometry, pages 111–144. Oxford Univ. Press, Oxford, 2010. MR2654677
Yukich, J. E.: Surface order scaling in stochastic geometry. Ann. Appl. Probab., 25, (2015), 177–210. MR3297770
