Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-28T13:31:37.080Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost sure central limit theorems in stochastic geometry

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  24 September 2020

Giovanni Luca Torrisi  and
Emilio Leonardi
Giovanni Luca Torrisi*
Affiliation:
Istituto per le Applicazioni del Calcolo, CNR
Emilio Leonardi*
Affiliation:
Politecnico di Torino
*
*Postal address: Via dei Taurini 19, 00185 Roma, Italy. Email: giovanniluca.torrisi@cnr.it
**Postal address: Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Email: emilio.leonardi@polito.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of the k-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation.

Keywords

Almost sure limit theoremMalliavin calculusPoisson processrandom graphsstabilizationstochastic geometry

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 705 - 734
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York.Google Scholar
Azmoodeh, E. and Nourdin, I. (2019). Almost sure limit theorems on Wiener chaos: the non-central case. Electron. Commun. Prob. 24, 1–12.CrossRefGoogle Scholar
Bercu, B., Nourdin, I. and Taqqu, M. S. (2010). Almost sure central limit theorems on the Wiener space. Stoch. Proc. Appl. 120, 1607–1628.CrossRefGoogle Scholar
Berkes, I. and Csáki, E. (2001). A universal result in almost sure central limit theory. Stoch. Proc. Appl. 94, 105–134.CrossRefGoogle Scholar
Brosamler, G. (1988). An almost everywhere central limit theorem. Math. Proc. Camb. Phil. Soc. 104, 561–574.CrossRefGoogle Scholar
Chazottes, J. R. and Gouëzel, S. (2007). On almost sure versions of classical limit theorems for dynamical systems. 138, 195–234.CrossRefGoogle Scholar
Ibragimov, I. A. and Lifshits, M. A. (2000). On limit theorems of almost sure type. 44, 254–272.Google Scholar
Lachièze-Rey, R., Schulte, M. and Yukich, J. E. (2019). Normal approximation for stabilizing functionals. Ann. Appl. Prob. 29, 931–993.CrossRefGoogle Scholar
Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. 150, 663–690.CrossRefGoogle Scholar
Last, G. and Penrose, M. D. (2017). Lectures on the Poisson Process. Cambridge University Press.CrossRefGoogle Scholar
Last, G., Peccati, G. and Schulte, M. (2014). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. 165, 667–723.Google Scholar
Lévy, P. (1937). Théorie de l’addition des variables aléatoires. Gauthier-Villar, Paris.Google Scholar
Peligrad, M. and Shao, Q. M. (1995). A note on the almost sure central limit theorem for weakly dependent random variables. Statist. Prob. Lett. 22, 131–136.CrossRefGoogle Scholar
Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press, New York.CrossRefGoogle Scholar
Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.CrossRefGoogle 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. (2013). Limit theory for point processes in manifolds. Ann. Appl. Prob. 23, 2161–2211.CrossRefGoogle Scholar
Reitzner, M., Spodarev, E. and Zhaporazhets, D. (2012). Set reconstruction by Voronoi cells. 44, 938–953.Google Scholar
Schatte, P. (1988). On strong versions of the central limit theorem. 137, 249–256.CrossRefGoogle Scholar
Schulte, M. (2012). A central limit theorem for the Poisson–Voronoi approximation. Adv. Appl. Math. 49, 285–306.CrossRefGoogle Scholar
Thäle, C. and Yukich, J. E. (2016). Asymptotic theory for statistics of the Poisson–Voronoi approximation. Bernoulli 22, 2372–2400.CrossRefGoogle Scholar
Yoshihara, K. I. (2004). Weakly dependent stochastic sequences and their applications. In Recent Topics on Weak and Strong Limit Theorems, Vol. XIV, Sanseido, Chiyoda.Google Scholar
Yukich, J. E. (2015). Surface order scaling in stochastic geometry. Ann. Appl. Prob. 25, 177–210.CrossRefGoogle Scholar
Zheng, C. (2011). Multi-dimensional Malliavin–Stein method on the Poisson space and its applications to limit theorems. Doctoral Thesis, Université Pierre et Marie Curie. Available at https://tel.archives-ouvertes.fr/tel-00648611/document.Google Scholar
Zheng, G. (2017). Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals. Stoch. Proc. Appl. 127, 1622–1636.CrossRefGoogle Scholar