This article presents a complete second-order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results do not require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case, the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.
Ann. Appl. Probab.
29(5):
2613-2653
(October 2019).
DOI: 10.1214/18-AAP1445
[1] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs 254. Oxford University Press, New York. MR1857292 0957.49001[1] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs 254. Oxford University Press, New York. MR1857292 0957.49001
[2] Baccelli, F. and Biswas, A. (2015). On scaling limits of power law shot-noise fields. Stoch. Models 31 187–207. MR3344212 1328.60024 10.1080/15326349.2014.990980[2] Baccelli, F. and Biswas, A. (2015). On scaling limits of power law shot-noise fields. Stoch. Models 31 187–207. MR3344212 1328.60024 10.1080/15326349.2014.990980
[4] 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. 0502.62045 10.1214/aop/1176993668 euclid.aop/1176993668[4] 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. 0502.62045 10.1214/aop/1176993668 euclid.aop/1176993668
[5] Biermé, H. and Desolneux, A. (2012). Crossings of smooth shot noise processes. Ann. Appl. Probab. 22 2240–2281. MR3024968 1278.60073 10.1214/11-AAP807 euclid.aoap/1353695953[5] Biermé, H. and Desolneux, A. (2012). Crossings of smooth shot noise processes. Ann. Appl. Probab. 22 2240–2281. MR3024968 1278.60073 10.1214/11-AAP807 euclid.aoap/1353695953
[6] Biermé, H. and Desolneux, A. (2016). Mean geometry for 2d random fields: Level perimeter and level total curvature integrals. Preprint HAL, No. 01370902.[6] Biermé, H. and Desolneux, A. (2016). Mean geometry for 2d random fields: Level perimeter and level total curvature integrals. Preprint HAL, No. 01370902.
[7] Biermé, H. and Desolneux, A. (2016). On the perimeter of excursion sets of shot noise random fields. Ann. Probab. 44 521–543. 1343.60060 10.1214/14-AOP980 euclid.aop/1454423048[7] Biermé, H. and Desolneux, A. (2016). On the perimeter of excursion sets of shot noise random fields. Ann. Probab. 44 521–543. 1343.60060 10.1214/14-AOP980 euclid.aop/1454423048
[8] Biermé, H., Di Bernardino, E., Duval, C. and Estrade, A. (2019). Lipschitz–Killing curvatures of excursion sets for two-dimensional random fields. Electron. J. Stat. 13 536–581. 1406.60076 10.1214/19-EJS1530[8] Biermé, H., Di Bernardino, E., Duval, C. and Estrade, A. (2019). Lipschitz–Killing curvatures of excursion sets for two-dimensional random fields. Electron. J. Stat. 13 536–581. 1406.60076 10.1214/19-EJS1530
[9] Bulinski, A., Spodarev, E. and Timmermann, F. (2012). Central limit theorems for the excursion set volumes of weakly dependent random fields. Bernoulli 18 100–118. 1239.60017 10.3150/10-BEJ339 euclid.bj/1327068619[9] Bulinski, A., Spodarev, E. and Timmermann, F. (2012). Central limit theorems for the excursion set volumes of weakly dependent random fields. Bernoulli 18 100–118. 1239.60017 10.3150/10-BEJ339 euclid.bj/1327068619
[11] Estrade, A. and León, J. R. (2016). A central limit theorem for the Euler characteristic of a Gaussian excursion set. Ann. Probab. 44 3849–3878. 1367.60016 10.1214/15-AOP1062 euclid.aop/1479114264[11] Estrade, A. and León, J. R. (2016). A central limit theorem for the Euler characteristic of a Gaussian excursion set. Ann. Probab. 44 3849–3878. 1367.60016 10.1214/15-AOP1062 euclid.aop/1479114264
[12] Galerne, B. and Lachièze-Rey, R. (2015). Random measurable sets and covariogram realizability problems. Adv. in Appl. Probab. 47 611–639. 1353.60013 10.1017/S0001867800048758 euclid.aap/1444308874[12] Galerne, B. and Lachièze-Rey, R. (2015). Random measurable sets and covariogram realizability problems. Adv. in Appl. Probab. 47 611–639. 1353.60013 10.1017/S0001867800048758 euclid.aap/1444308874
[15] Lachièze-Rey, R. and Peccati, G. (2017). New Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 1992–2031. 1374.60023 10.1214/16-AAP1218 euclid.aoap/1504080024[15] Lachièze-Rey, R. and Peccati, G. (2017). New Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 1992–2031. 1374.60023 10.1214/16-AAP1218 euclid.aoap/1504080024
[16] Lachièze-Rey, R., Schulte, M. and Yukich, J. E. (2019). Normal approximation for stabilizing functionals. Ann. Appl. Probab. 29 931–993. 07047442 10.1214/18-AAP1405 euclid.aoap/1548298934[16] Lachièze-Rey, R., Schulte, M. and Yukich, J. E. (2019). Normal approximation for stabilizing functionals. Ann. Appl. Probab. 29 931–993. 07047442 10.1214/18-AAP1405 euclid.aoap/1548298934
[17] Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin. 0990.86007[17] Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin. 0990.86007
[18] Last, G., Peccati, G. and Schulte, M. (2016). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields 165 667–723. 1347.60012 10.1007/s00440-015-0643-7[18] Last, G., Peccati, G. and Schulte, M. (2016). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields 165 667–723. 1347.60012 10.1007/s00440-015-0643-7
[19] Levina, E. and Bickel, P. J. (2005). Maximum likelihood estimation of intrinsic dimension. In Advances in Neural Information Processing Systems 777–784.[19] Levina, E. and Bickel, P. J. (2005). Maximum likelihood estimation of intrinsic dimension. In Advances in Neural Information Processing Systems 777–784.
[20] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041. 1044.60016 euclid.aoap/1015345393[20] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041. 1044.60016 euclid.aoap/1015345393
[21] Rataj, J. and Winter, S. (2010). On volume and surface area of parallel sets. Indiana Univ. Math. J. 59 1661–1685. 1234.28008 10.1512/iumj.2010.59.4165[21] Rataj, J. and Winter, S. (2010). On volume and surface area of parallel sets. Indiana Univ. Math. J. 59 1661–1685. 1234.28008 10.1512/iumj.2010.59.4165
[22] Reddy, T. R., Vadlamani, S. and Yogeshwaran, D. (2018). Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs. J. Stat. Phys. 173 941–984. 1405.82011 10.1007/s10955-018-2026-9[22] Reddy, T. R., Vadlamani, S. and Yogeshwaran, D. (2018). Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs. J. Stat. Phys. 173 941–984. 1405.82011 10.1007/s10955-018-2026-9
[23] Schlather, M., Ribeiro, P. J. Jr. and Diggle, P. J. (2004). Detecting dependence between marks and locations of marked point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 79–93. 1061.62151 10.1046/j.1369-7412.2003.05343.x[23] Schlather, M., Ribeiro, P. J. Jr. and Diggle, P. J. (2004). Detecting dependence between marks and locations of marked point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 79–93. 1061.62151 10.1046/j.1369-7412.2003.05343.x
[24] Seppäläinen, T. and Yukich, J. E. (2001). Large deviation principles for Euclidean functionals and other nearly additive processes. Probab. Theory Related Fields 120 309–345. 0984.60038 10.1007/PL00008785[24] Seppäläinen, T. and Yukich, J. E. (2001). Large deviation principles for Euclidean functionals and other nearly additive processes. Probab. Theory Related Fields 120 309–345. 0984.60038 10.1007/PL00008785