A limit theorem for persistence diagrams of random filtered complexes built over marked point processes
Volume 10, Issue 1 (2023), pp. 1–18
Pub. online: 13 September 2022
Type: Research Article
Open Access
Open Access
Received
4 February 2022
4 February 2022
Revised
9 August 2022
9 August 2022
Accepted
29 August 2022
29 August 2022
Published
13 September 2022
13 September 2022
Abstract
Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $\check{\text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.
References
Baccelli, F., Blaszczyszyn, B.: Stochastic Geometry and Wireless Networks, Volume I – Theory. Found. Trends Netw. 3(3-4), 249–449, (2009). Vol. 1, pp. 150. NoW Publishers. https://doi.org/10.1561/1300000006, https://hal.inria.fr/inria-00403039. Stochastic Geometry and Wireless Networks, Volume II – Applications; see http://hal.inria.fr/inria-00403040.
Bell, G., Lawson, A., Martin, J., Rudzinski, J., Smyth, C.: Weighted persistent homology. Involve 12(5), 823–837 (2019). MR3954298. https://doi.org/10.2140/involve.2019.12.823
Bobrowski, O., Mukherjee, S.: The topology of probability distributions on manifolds. Probab. Theory Related Fields 161(3–4), 651–686 (2015). MR3334278. https://doi.org/10.1007/s00440-014-0556-x
Bobrowski, O., Oliveira, G.: Random Čech complexes on Riemannian manifolds. Random Structures Algorithms 54(3), 373–412 (2019). MR3938773. https://doi.org/10.1002/rsa.20800
Buchet, M., Chazal, F., Oudot, S.Y., Sheehy, D.R.: Efficient and robust persistent homology for measures. Comput. Geom. 58, 70–96 (2016). MR3541079. https://doi.org/10.1016/j.comgeo.2016.07.001
Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications, 3rd edn. Wiley Series in Probability and Statistics, p. 544. John Wiley & Sons, Ltd., Chichester (2013). MR3236788. https://doi.org/10.1002/9781118658222
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd edn. Probability and its Applications (New York), p. 469. Springer, (2003). MR1950431
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd edn. Probability and its Applications (New York), p. 573. Springer, (2008). MR2371524. https://doi.org/10.1007/978-0-387-49835-5
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Discrete and computational geometry and graph drawing, Columbia, SC, 2001, vol. 28, pp. 511–533 (2002). MR1949898. https://doi.org/10.1007/s00454-002-2885-2
Fritz, J.: Generalization of McMillan’s theorem to random set functions. Studia Sci. Math. Hungar. 5, 369–394 (1970). MR293956
Goel, A., Trinh, K.D., Tsunoda, K.: Strong law of large numbers for Betti numbers in the thermodynamic regime. J. Stat. Phys. 174(4), 865–892 (2019). MR3913900. https://doi.org/10.1007/s10955-018-2201-z
Hiraoka, Y., Shirai, T., Trinh, K.D.: Limit theorems for persistence diagrams. Ann. Appl. Probab. 28(5), 2740–2780 (2018). MR3847972. https://doi.org/10.1214/17-AAP1371
Meng, Z., Anand, D.V., Lu, Y., Wu, J., Xia, K.: Weighted persistent homology for biomolecular data analysis (2020). https://doi.org/10.1038/s41598-019-55660-3
Nguyen, T.V., Baccelli, F.: On the Generating Functionals of a Class of Random Packing Point Processes (2013). arXiv: 1311.4967. https://doi.org/10.48550/ARXIV.1311.4967
Nguyen, X.-X., Zessin, H.: Ergodic theorems for spatial processes. Z. Wahrsch. Verw. Gebiete 48(2), 133–158 (1979). MR534841. https://doi.org/10.1007/BF01886869
Pugh, C., Shub, M.: Ergodic elements of ergodic actions. Compos. Math. 23, 115–122 (1971). MR283174
Spitz, D., Wienhard, A.: The self-similar evolution of stationary point processes via persistent homology (2020). arXiv: 2012.05751. https://doi.org/10.48550/ARXIV.2012.05751
Xuan-Xanh, N., Zessin, H.: Punktprozesse mit Wechselwirkung. Z. Wahrsch. Verw. Gebiete 37(2), 91–126 (1976/77). MR423601. https://doi.org/10.1007/BF00536775
Yogeshwaran, D., Adler, R.J.: On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25(6), 3338–3380 (2015). MR3404638. https://doi.org/10.1214/14-AAP1075
Yogeshwaran, D., Subag, E., Adler, R.J.: Random geometric complexes in the thermodynamic regime. Probab. Theory Related Fields 167(1–2), 107–142 (2017). MR3602843. https://doi.org/10.1007/s00440-015-0678-9
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005). MR2121296. https://doi.org/10.1007/s00454-004-1146-y
