The Annals of Probability

Limit theorems for point processes under geometric constraints (and topological crackle)

Takashi Owada and Robert J. Adler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered.

The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological “crackle;” the continued presence of spurious homology in samples of topological structures, despite increased sample size.

Article information

Ann. Probab., Volume 45, Number 3 (2017), 2004-2055.

Received: March 2015
Revised: January 2016
First available in Project Euclid: 15 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 55U10: Simplicial sets and complexes

Point process Poisson random measure extreme value theory regular variation Čech complex Betti number topological data analysis crackle geometric graph


Owada, Takashi; Adler, Robert J. Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 (2017), no. 3, 2004--2055. doi:10.1214/16-AOP1106.

Export citation


  • [1] Adler, R. J., Bobrowski, O. and Weinberger, S. (2014). Crackle: The homology of noise. Discrete Comput. Geom. 52 680–704.
  • [2] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  • [3] Balkema, G., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101 1738–1754.
  • [4] Balkema, G. and Embrechts, P. (2004). Multivariate excess distributions. Available at
  • [5] Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society (EMS), Zürich.
  • [6] Balkema, G., Embrechts, P. and Nolde, N. (2013). The shape of asymptotic dependence. In Prokhorov and Contemporary Probability Theory. Springer Proc. Math. Stat. 33 43–67. Springer, Heidelberg.
  • [7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [8] Bobrowski, O. and Adler, R. J. (2014). Distance functions, critical points, and the topology of random Čech complexes. Homology Homotopy Appl. 16 311–344.
  • [9] Borsuk, K. (1948). On the imbedding of systems of compacta in simplicial complexes. Fund. Math. 35 217–234.
  • [10] Dabrowski, A. R., Dehling, H. G., Mikosch, T. and Sharipov, O. (2002). Poisson limits for $U$-statistics. Stochastic Process. Appl. 99 137–157.
  • [11] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Springer, New York.
  • [12] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195.
  • [13] Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879–917.
  • [14] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [16] Hatcher, A. (2002). Algebraic Topology. Cambridge Univ. Press, Cambridge.
  • [17] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
  • [18] Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology Homotopy Appl. 15 343–374.
  • [19] Niyogi, P., Smale, S. and Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39 419–441.
  • [20] Niyogi, P., Smale, S. and Weinberger, S. (2011). A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40 646–663.
  • [21] Owada, T. and Samorodnitsky, G. (2015). Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21 1575–1599.
  • [22] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [23] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [24] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • [25] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • [26] Schulte, M. and Thäle, C. (2012). The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stochastic Process. Appl. 122 4096–4120.
  • [27] Vick, J. W. (1994). Homology Theory: An Introduction to Algebraic Topology, 2nd ed. Graduate Texts in Mathematics 145. Springer, New York.
  • [28] Yogeshwaran, D. and Adler, R. J. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 3338–3380.
  • [29] Yogeshwaran, D., Subag, E. and Adler, R. J. (2016). Random geometric complexes in the thermodynamic regime. Probab. Theory Related Fields. To appear. Available at arXiv:1403.1164.