Project Euclid

References

• [1] Adler, R. J., Bobrowski, O., Borman, M. S., Subag, E. and Weinberger, S. (2010). Persistent homology for random fields and complexes. In Borrowing Strength: Theory Powering Applications—a Festschrift for Lawrence D. Brown. Inst. Math. Stat. (IMS) Collect. 6 124–143. IMS, Beachwood, OH.
• [2] Adler, R. J., Bobrowski, O. and Weinberger, S. (2014). Crackle: The homology of noise. Discrete Comput. Geom. 52 680–704.
  Zentralblatt MATH: 1350.60010
  Digital Object Identifier: doi:10.1007/s00454-014-9621-6
  
• [3] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  Zentralblatt MATH: 0675.60017
  Digital Object Identifier: doi:10.1214/aop/1176991491
  Project Euclid: euclid.aop/1176991491
  
• [4] Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society (EMS), Zürich.
  Zentralblatt MATH: 1121.91055
  
• [5] Balkema, G., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101 1738–1754.
  Zentralblatt MATH: 1198.60012
  Digital Object Identifier: doi:10.1016/j.jmva.2010.02.010
  
• [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.
  Zentralblatt MATH: 1274.60165
  
• [7] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  Zentralblatt MATH: 0172.21201
  
• [8] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  Zentralblatt MATH: 0944.60003
  
• [9] Björner, A. (1995). Topological methods. In Handbook of Combinatorics, Vols 1, 2 1819–1872. Elsevier, Amsterdam.
• [10] Bobrowski, O. and Kahle, M. (2014). Topology of random simplicial complexes: A survey. In Algebraic Topology: Applications and New Directions. Contemp. Math. 620 201–221. Amer. Math. Soc., Providence, RI.
  Zentralblatt MATH: 1333.05324
  
• [11] Bobrowski, O., Kahle, M. and Skraba, P. (2017). Maximally persistent cycles in random geometric complexes. Ann. Appl. Probab. 27 2032–2060.
  Zentralblatt MATH: 1377.60024
  Digital Object Identifier: doi:10.1214/16-AAP1232
  Project Euclid: euclid.aoap/1504080025
  
• [12] Bobrowski, O. and Mukherjee, S. (2015). The topology of probability distributions on manifolds. Probab. Theory Related Fields 161 651–686.
  Zentralblatt MATH: 1316.60020
  Digital Object Identifier: doi:10.1007/s00440-014-0556-x
  
• [13] Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16 77–102.
  Zentralblatt MATH: 1337.68221
  
• [14] Carlsson, G. (2014). Topological pattern recognition for point cloud data. Acta Numer. 23 289–368.
• [15] Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  Zentralblatt MATH: 0980.62043
  
• [16] Dabaghian, Y., Memoli, F., Frank, L. and Carlsson, G. (2012). A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput. Biol. 8 e1002581.
• [17] Dabrowski, A. R., Dehling, H. G., Mikosch, T. and Sharipov, O. (2002). Poisson limits for $U$-statistics. Stochastic Process. Appl. 99 137–157.
  Zentralblatt MATH: 1059.60063
  Digital Object Identifier: doi:10.1016/S0304-4149(01)00153-3
  
• [18] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  Zentralblatt MATH: 1101.62002
  
• [19] de Silva, V. and Ghrist, R. (2007). Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7 339–358.
  Zentralblatt MATH: 1134.55003
  Digital Object Identifier: doi:10.2140/agt.2007.7.339
  Project Euclid: euclid.agt/1513796672
  
• [20] Decreusefond, L., Schulte, M. and Thäle, C. (2016). Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Probab. 44 2147–2197.
  Zentralblatt MATH: 1347.60027
  Digital Object Identifier: doi:10.1214/15-AOP1020
  Project Euclid: euclid.aop/1463410041
  
• [21] Duy, T. K., Hiraoka, Y. and Shirai, T. (2016). Limit theorems for persistence diagrams. Available at arXiv:1612.08371.
  arXiv: 1612.08371
  
• [22] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction. Amer. Math. Soc., Providence, RI.
  Zentralblatt MATH: 1193.55001
  
• [23] Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Topological persistence and simplification. Discrete Comput. Geom. 28 511–533.
  Zentralblatt MATH: 1011.68152
  Digital Object Identifier: doi:10.1007/s00454-002-2885-2
  
• [24] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
• [25] Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S. and Singh, A. (2014). Confidence sets for persistence diagrams. Ann. Statist. 42 2301–2339.
  Zentralblatt MATH: 1310.62059
  Digital Object Identifier: doi:10.1214/14-AOS1252
  Project Euclid: euclid.aos/1413810729
  
• [26] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Melbourne, FL.
  Zentralblatt MATH: 0634.62044
  
• [27] Ghrist, R. (2008). Barcodes: The persistent topology of data. Bull. Amer. Math. Soc. (N.S.) 45 61–75.
  Zentralblatt MATH: 1391.55005
  Digital Object Identifier: doi:10.1090/S0273-0979-07-01191-3
  
• [28] Ghrist, R. (2014). Elementary Applied Topology, Createspace.
• [29] Hatcher, A. (2002). Algebraic Topology. Cambridge Univ. Press, Cambridge.
  Mathematical Reviews (MathSciNet): MR1867354
  Zentralblatt MATH: 1044.55001
  
• [30] Hiraoka, Y. and Shirai, T. (2017). Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process. Random Structures Algorithms 51 315–340.
  Zentralblatt MATH: 1370.05190
  Digital Object Identifier: doi:10.1002/rsa.20718
  
• [31] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
  Zentralblatt MATH: 1219.05175
  Digital Object Identifier: doi:10.1007/s00454-010-9319-3
  
• [32] Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology, Homotopy Appl. 15 343–374.
  Zentralblatt MATH: 1268.05180
  Digital Object Identifier: doi:10.4310/HHA.2013.v15.n1.a17
  Project Euclid: euclid.hha/1383943681
  
• [33] Kahle, M. and Meckes, E. (2016). Erratum to “Limit theorems for Betti numbers of random simplicial complexes.” Homology, Homotopy Appl. 18 129–142.
  Zentralblatt MATH: 06618792
  Digital Object Identifier: doi:10.4310/HHA.2016.v18.n1.a7
  
• [34] Kusano, G., Fukumizu, K. and Hiraoka, Y. (2016). Persistence weighted Gaussian kernel for topological data analysis. In ICML’16 Proceedings of the 33rd International Conference on International Conference on Machine Learning, Vol. 48, pp. 2004–2013.
• [35] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York–Berlin.
• [36] Martin, S., Thompson, A., Coutsias, E. A. and Watson, J. (2010). Topology of cyclo-octane energy landscape. J. Chem. Phys. 132 234115.
• [37] Munkres, J. R. (1984). Elements of Algebraic Topology, 1st ed. Benjamin-Cummings, San Francisco, CA.
  Zentralblatt MATH: 0673.55001
  
• [38] 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.
  Zentralblatt MATH: 1148.68048
  Digital Object Identifier: doi:10.1007/s00454-008-9053-2
  
• [39] Owada, T. (2017). Functional central limit theorem for subgraph counting processes. Electron. J. Probab. 22 Paper No. 17, 38.
  Zentralblatt MATH: 1357.60056
  Digital Object Identifier: doi:10.1214/17-EJP30
  
• [40] Owada, T. and Adler, R. J. (2017). Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 2004–2055.
  Zentralblatt MATH: 1373.60090
  Digital Object Identifier: doi:10.1214/16-AOP1106
  Project Euclid: euclid.aop/1494835236
  
• [41] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [42] Port, A., Gheorghita, I., Guth, D., Clark, J. M., Liang, C., Dasu, S. and Marcolli, M. (2015). Persistent topology of syntax. Available at arXiv:1507.05134.
  arXiv: 1507.05134
  
• [43] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  Zentralblatt MATH: 0633.60001
  
• [44] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  Zentralblatt MATH: 1152.62029
  
• [45] 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.
  Zentralblatt MATH: 1262.60031
  Digital Object Identifier: doi:10.1016/j.spa.2012.08.011
  
• [46] Vick, J. W. (1994). Homology Theory: An Introduction to Algebraic Topology, 2nd ed. Graduate Texts in Mathematics 145. Springer, New York.
  Zentralblatt MATH: 0789.55004
  
• [47] Yogeshwaran, D. and Adler, R. J. (2015). On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 3338–3380.
  Zentralblatt MATH: 1328.60123
  Digital Object Identifier: doi:10.1214/14-AAP1075
  Project Euclid: euclid.aoap/1443703776
  
• [48] Yogeshwaran, D., Subag, E. and Adler, R. J. (2017). Random geometric complexes in the thermodynamic regime. Probab. Theory Related Fields 167 107–142.
  Zentralblatt MATH: 1366.60033
  Digital Object Identifier: doi:10.1007/s00440-015-0678-9
  
• [49] Zomorodian, A. and Carlsson, G. (2005). Computing persistent homology. Discrete Comput. Geom. 33 249–274.
  Zentralblatt MATH: 1069.55003
  Digital Object Identifier: doi:10.1007/s00454-004-1146-y