Increasing domain asymptotics for the first Minkowski functional of spherical random fields
Authors:
N. N. Leonenko and M. D. Ruiz-Medina
Journal:
Theor. Probability and Math. Statist. 97 (2018), 127-149
MSC (2010):
Primary 60G60, 60F05, 60G10, 62E20
DOI:
https://doi.org/10.1090/tpms/1053
Published electronically:
February 21, 2019
MathSciNet review:
3746004
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Additional Information
Abstract: The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in $\mathbb {R}^{3}$. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation.
References
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References
- R. Adler and J. Taylor, Random Fields and Geometry, Springer, New York, 2007. MR 2319516
- S. M. Berman, Sojourns of vector Gaussian processes inside and outside spheres, Z. Wahrsch. Verw. Gebiete 66 (1984), 529–542. MR 753812
- H. Dehling and M. S. Taqqu, The empirical process of some long-range dependent sequences with an application to U-statistics, Ann. Statist. 17 (1989), 1767–1783. MR 1026312
- R. L. Dobrushin and P. Major, Non-central limit theorem for non-linear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), 1–28. MR 550122
- C. Hikage, E. Komatsu, and T. Matsubara, Primordial non-Gaussianity and analytical formula for Minkowski functionals of the cosmic microwave background and large-scale structure, The Astrophysical Journal 653 (2006), 11–26.
- A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer Academic Publishers, Dordrecht, 1989. MR 1009786
- A. Lang and C. Schwab, Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab. 25 (2015), 3047–3094. MR 3404631
- N. N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum, Kluwer Academic Publishers, Dordrecht, 1999. MR 1687092
- N. N. Leonenko and A. Ya. Olenko, Tauberian theorems for correlation functions and limit theorems for spherical averages of random fields, Random Oper. Stoch. Equ. 1 (1993), 57–67. MR 1254176
- N. Leonenko and A. Olenko, Tauberian and Abelian theorems for long-range dependent random fields, Methodol. Comput. Appl. Probab. 15 (2013), 715–742. MR 3117624
- N. Leonenko and A. Olenko, Sojourn measures of Student and Fisher–Snedecor random fields, Bernoulli 20 (2014), 1454–1483. MR 3217450
- N. Leonenko, M. D. Ruiz-Medina, and M. S. Taqqu, Non-central limit theorems for random fields subordinated to gamma-correlated random fields, Bernoulli 23 (2017), 3469–3507. MR 3654813
- N. N. Leonenko, M. D. Ruiz-Medina, and M. S. Taqqu, Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence, Stoch. Anal. Appl. 35 (2017), 144–177. MR 3581700
- N. N. Leonenko and K. V. Rybasov, Conditions for convergence to a Wiener process of spherical means of local functionals of Gaussian fields, Theory Probab. Math. Statist. 34 (1986), 85–93. MR 887927
- N. N. Leonenko and Sh. O. Sabirov, Spherical measures for the exceedence of a level for a class of random fields, Cybernetics 25 (1989), 272–280. MR 1009707
- N. N. Leonenko, M. S. Taqqu, and G. Terdik, Estimation of the covariance function of Gaussian isotropic random fields on spheres, related Rosenblatt-type distributions and the cosmic variance problem, Electron. J. Stat. 12 (2018), no. 2, 3114–3146. MR 3857874
- E. Lukacs, Characteristic Functions, 2nd ed., Griffin, London, 1970. MR 0124075
- P. Major, Multiple Wiener–Itô Integrals: With Applications to Limit Theorems, Lecture Notes in Math., Springer, New York, 1981. MR 611334
- D. Marinucci and G. Peccati, Random Fields on the Sphere. Representation, Limit Theorems and Cosmological Applications, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011. MR 2840154
- D. Marinucci and G. Peccati, Mean-square continuity on homogeneous spaces of compact groups, Electron. Commun. Probab. 18 (2013), 1–10. MR 3064996
- D. Munshi, J. Smidt, A. Cooray, A. Renzi, A. Heavens, and P. Coles, New approaches to probing Minkowski functionals, Monthly Notices of the Royal Astronomical Society 434 (2013), 2830–2855.
- G. Peccati and M. S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams, Springer, New York, 2011. MR 2791919
- V. H. Pham, On the rate of convergence for central limit theorems of sojourn times of Gaussian fields, Stoch. Process. Appl. 123 (2013), 2158–2174. MR 3038501
- V. H. Pham, Quantitative central limit theorems of sojourn times of isotropic Gaussian fields, Acta Math. Vietnam 42 (2017), 637–651. MR 3708033
- Planck 2015 results. XVI, Isotropy and statistics of the CMB, Planck Collaboration, Astronomy and Astrophysics 594 (2016), A16.
- Planck 2015 results. I, Overview of products and scientific results, Astronomy and Astrophysics 594 (2016), A16.
- Yu. D. Popov and M. Yadrenko, Certain questions of the spectral theory of homogeneous and isotropic random fields, Theory Probab. Appl. 14 (1969), 531–540. MR 0278374
- M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, 1980. MR 751959
- J. Schmalzing and K. M. Gorski, Minkowski functionals used in the morphological analysis of Cosmic Microwave Background anisotropy maps, Mon. Not. R. Astron. Soc. 297 (1998), 355–365.
- E. M. Stein, Singular Integrals and Differential Properties of Functions, Princenton University Press, New Jersey, 1970. MR 0290095
- M. S. Taqqu, Weak-convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. Verw. Gebiete 31 (1975), 287–302. MR 0400329
- M. S. Veillette and M. S. Taqqu, Properties and numerical evaluation of Rosenblatt distribution, Bernoulli, 19 (2013), 982–1005. MR 3079303
- M. I. Yadrenko, Spectral Theory of Random Fields, Optimization Software Inc., New York, 1983. MR 697386
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Additional Information
N. N. Leonenko
Affiliation:
Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, United Kingdom
Email:
LeonenkoN@cardiff.ac.uk
M. D. Ruiz-Medina
Affiliation:
Department of Statistics and Operations Research, Campus de Fuente Nueva s/n, University of Granada, E-18071 Granada, Spain
Email:
mruiz@ugr.es
Keywords:
Central and noncentral limit theorems,
chi-squared random fields,
Gaussian random fields,
Karhunen–Loéve expansion,
spherical Rosenblatt distribution
Received by editor(s):
September 17, 2017
Published electronically:
February 21, 2019
Additional Notes:
The first author was supported in particular by Cardiff Incoming Visiting Fellowship Scheme, International Collaboration Seedcorn Fund, Data Innovation URI Seedcorn Fund, and Australian Research Council’s Discovery Projects funding scheme (project DP160101366).
The authors were supported by project MTM2015-71839-P of MINECO, Spain (co-funded with FEDER funds).
Dedicated:
This contribution is dedicated to the 85th anniversary of Professor Mykhailo Yosypovych Yadrenko
Article copyright:
© Copyright 2019
American Mathematical Society