The Annals of Applied Probability

Limit theory for point processes in manifolds

Mathew D. Penrose and J. E. Yukich

Full-text: Open access

Abstract

Let $Y_{i}$, $i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset\mathbb{R}^{d}$ and consider sums $\sum_{i=1}^{n}\xi(n^{1/m}Y_{i},\{n^{1/m}Y_{j}\}_{j=1}^{n})$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal{Y} )$, with $y\in\mathbb{R}^{d}$ and $\mathcal{Y}\subset\mathbb{R}^{d}$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal{M} $. We apply the general results to establish the limit theory of dimension and volume content estimators, Rényi and Shannon entropy estimators and clique counts in the Vietoris–Rips complex on $\{Y_{i}\}_{i=1}^{n}$.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2161-2211.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447685

Digital Object Identifier
doi:10.1214/12-AAP897

Mathematical Reviews number (MathSciNet)
MR3127932

Zentralblatt MATH identifier
1285.60021

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Manifolds dimension estimators entropy estimators Vietoris–Rips complex clique counts

Citation

Penrose, Mathew D.; Yukich, J. E. Limit theory for point processes in manifolds. Ann. Appl. Probab. 23 (2013), no. 6, 2161--2211. doi:10.1214/12-AAP897. https://projecteuclid.org/euclid.aoap/1382447685


Export citation

References

  • [1] Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247–258.
  • [2] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033–1046.
  • [3] Baryshnikov, Y., Penrose, M. D. and Yukich, J. E. (2008). Gaussian limits for generalized spacings. Extended version. Available at arXiv:0804.4123v1.
  • [4] Baryshnikov, Y., Penrose, M. D. and Yukich, J. E. (2009). Gaussian limits for generalized spacings. Ann. Appl. Probab. 19 158–185.
  • [5] Baryshnikov, Y. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 213–253.
  • [6] Beirlant, J., Dudewicz, E., Györfi, L. and van der Meulen, E. (1997). Non-parametric entropy estimation: An overview. Int. J. Math. Statist. Sci. 6 17–39.
  • [7] Berger, M. and Gostiaux, B. (1988). Differential Geometry: Manifolds, Curves, and Surfaces. Graduate Texts in Mathematics 115. Springer, New York.
  • [8] Bhattacharya, R. N. and Ghosh, J. K. (1992). A class of $U$-statistics and asymptotic normality of the number of $k$-clusters. J. Multivariate Anal. 43 300–330.
  • [9] Bickel, P. J. and Breiman, L. (1983). Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 185–214.
  • [10] Bickel, P. J. and Yan, D. (2008). Sparsity and the possibility of inference. Sankhyā 70 1–24.
  • [11] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [12] Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46 255–308.
  • [13] Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
  • [14] Chazal, F., Guibas, L. J., Oudot, S. Y. and Skraba, P. (2009). Analysis of scalar fields over point cloud data. In Proceedings of the Twentieth Annual ACM–SIAM Symposium on Discrete Algorithms (C. Mathieu, ed.) 1021–1030. SIAM, Philadelphia, PA.
  • [15] Chazal, F. and Oudot, S. Y. (2008). Towards persistence-based reconstruction in Euclidean spaces. In Computational Geometry (SCG’08) (M. Teillaud, eds.) 232–241. ACM, New York.
  • [16] Costa, J. A. and Hero, A. O. III (2006). Determining intrinsic dimension and entropy of high-dimensional shape spaces. In Statistics and Analysis of Shapes (H. Krim and A. Yezzi, eds.) 231–252. Birkhäuser, Boston, MA.
  • [17] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • [18] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Springer, New York.
  • [19] de Silva, V. and Ghrist, R. (2006). Coordinate-free coverage in sensor networks with controlled boundaries via homology. Internat. J. Robotics Res. 25 1205–1222.
  • [20] de Silva, V. and Ghrist, R. (2007). Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7 339–358.
  • [21] Grassberger, P. and Procaccia, I. (1983). Measuring the strangeness of strange attractors. Phys. D 9 189–208.
  • [22] Havil, J. (2003). Gamma: Exploring Euler’s Constant. Princeton Univ. Press, Princeton, NJ.
  • [23] Havrda, J. and Charvát, F. (1967). Quantification method of classification processes. Concept of structural $a$-entropy. Kybernetika (Prague) 3 30–35.
  • [24] Jiménez, R. and Yukich, J. E. (2005). Statistical distances based on Euclidean graphs. In Recent Advances in Applied Probability (R. Baeza-Yates, J. Glaz, H. Gzyl, J. Hüsler and J. L. Palacios, eds.). 223–239. Springer, New York.
  • [25] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
  • [26] Kahle, M. and Meckes, E. (2010). Limit theorems for Betti numbers of random simplicial complexes. Unpublished manuscript. Available at arXiv:1009.4130v2. Homology, Homotopy Appl. To appear.
  • [27] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press Oxford Univ. Press, New York.
  • [28] Kozachenko, L. F. and Leonenko, N. N. (1987). Sample entropy of a random vector. Probl. Inf. Transm. 23 95–101.
  • [29] Leonenko, N., Pronzato, L. and Savani, V. (2008). A class of Rényi information estimators for multidimensional densities. Ann. Statist. 36 2153–2182.
  • [30] Levina, E. and Bickel, P. J. (2005). Maximum likelihood estimation of intrinsic dimension. In Advances in NIPS 17 (L. K. Saul, Y. Weiss and L. Bottou, eds.) 777–784. MIT Press, Cambridge.
  • [31] Liitiäinen, E., Lendasse, A. and Corona, F. (2010). A boundary corrected expansion of the moments of nearest neighbor distributions. Random Structures Algorithms 37 223–247.
  • [32] Nilsson, M. and Kleijn, W. B. (2004). Shannon entropy estimation based on high-rate quantization theory. In Proc. XII Europ. Signal Proc. Conf. (EUSIPCO) 1753–1756. Technishe Universität Wien, Vienna, Austria.
  • [33] Nilsson, M. and Kleijn, W. B. (2007). On the estimation of differential entropy from data located on embedded manifolds. IEEE Trans. Inform. Theory 53 2330–2341.
  • [34] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [35] Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. 12 989–1035 (electronic).
  • [36] Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 1124–1150.
  • [37] Penrose, M. D. and Yukich, J. E. (2011). Limit theory for point processes in manifolds. Available at arxiv:1104.0914v1.
  • [38] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041.
  • [39] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • [40] Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein’s Method and Applications (A. D. Barbour and L. H. Y. Chen, eds.) Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 37–58. Singapore Univ. Press, Singapore.
  • [41] Penrose, M. D. and Yukich, J. E. (2011). Laws of large numbers and nearest neighbor distances. In Advances in Directional and Linear Statistics (M. T. Wells and A. Sengupta, eds.) 189–199. Physica-Verlag/Springer, Heidelberg.
  • [42] Rényi, A. (1961). On measures of entropy and information. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 547–561. Univ. California Press, Berkeley, CA.
  • [43] Song, K.-S. (2001). Rényi information, loglikelihood and an intrinsic distribution measure. J. Statist. Plann. Inference 93 51–69.
  • [44] Wade, A. R. (2007). Explicit laws of large numbers for random nearest-neighbour-type graphs. Adv. in Appl. Probab. 39 326–342.
  • [45] Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.
  • [46] Yukich, J. E. (2008). Point process stabilization methods and dimension estimation. In Fifth Colloquium on Mathematics and Computer Science 59–69. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.