Estimates for Distributions of Suprema of Spherical Random Fields
Abstract
Bounds for distributions of suprema of $\varphi$-sub-Gaussian random fields defined over the $N$-dimensional unit sphere are stated. Applications of the results to the spherical fractional Brownian motion, isotropic Gaussian fields and some other models are presented.References
R.J. Adler, On excursion sets, tube formulas and maxima of random fields, Ann. Appl. Probab., vol. 10, no. 1, pp.
–74, 2000.
R.J. Adler, J.E. Taylor, Random Fields and Geometry, Springer Monographs in Mathematics, 448 p., 2007.
V. V. Buldygin, Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, American
Mathematical Society, Providence, RI, 257 p., 2000.
D. Cheng, P. Liu, Extremes of spherical fractional Brownian motion, Extremes, vol. 22, pp. 433–457, 2019.
D. Cheng, Y. Xiao, Excursion probability of Gaussian random fields on sphere, Bernoulli, vol. 22, no. 2, pp. 1113–1130,
I. Donhauzer, A. Olenko, Limit theorems for multifractal products of random fields, Preprint, arXiv:2202.02885, 2022.
M. D’Ovidio, E. Orsingher, L. Sakhno, Models of space-time random fields on the sphere, Modern Stoch. Theory
Appl., vol.9, iss. 2, pp. 139–156, 2022.
M. Dozzi, Yu. Kozachenko, Yu. Mishura, K. Ralchenko, Asymptotic growth of trajectories of multifractional Brownian
motion, with statistical applications to drift parameter estimation, Stat. Inference Stoch. Process., vol. 21, no. 1, pp. 21–52, 2018.
R. Giuliano Antonini, Yu.V. Kozachenko, T. Nikitina, Spaces of φ-subgaussian random variables, Rendiconti Accademia Nazionale delle Scienze XL. Memorie di Matematica e Applicazioni 121, vol. XXVII, pp. 95–124, 2003.
O. Hopkalo, Yu. Kozachenko, E. Orsingher, L. Sakhno, Sample Paths Properties of Stochastic Processes from Orlicz
Spaces, with Applications to Partial Differential Equations, Statistics Opt. Inform. Comput., vol. 8, no. 3, pp. 722–739,
O. Hopkalo, L. Sakhno, Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes, Modern Stoch. Theory Appl., vol. 8, iss.1, pp. 41–62, 2021.
J. Istas, Spherical and hyperbolic fractional Brownian motion, Electron. Commun. Probab., vol. 10, pp. 254–262,
J. Istas, Karhunen-Loève expansion of spherical fractional Brownian motions, Statist. Probab. Lett., vol. 76, pp. 1578–1583, 2006.
Yu.V. Kozachenko, Yu.A. Koval’chuk, Boundary value problems with random initial conditions and series of functions of Subφ(Ω), Ukrainian Math. J., vol. 50, no. 4, pp. 572–585, 1998.
Yu. Kozachenko, A. Olenko, Whitaker-Kotelnikov-Shanon approximation of φ-sub-Gaussian random processes, J. Math. Analysis Appl., vol. 442, no. 2, pp. 924–946, 2016.
Yu. Kozachenko, A. Olenko, Aliasing-truncation errors in sampling approximations of sub-gaussian signals, IEEE Trans. Inf. Theory., vol. 62, no. 10, pp. 5831–5838, 2016.
Yu. Kozachenko, A. Olenko, O. Polosmak, Convergence in Lp([0;t]) of wavelet expansions of φ-sub-gaussian random processes, Methodol. Comput. Appl. Probab., vol. 17, no.1, pp. 139–153, 2015.
Yu. Kozachenko, E. Orsingher, L. Sakhno, O. Vasylyk, Estimates for functional of solution to Higher-Order Heat-Type equation with random initial condition, J. Stat. Phys., vol. 72, no. 6, pp. 1641–1662, 2018.
Yu. Kozachenko, E. Orsingher, L. Sakhno, O. Vasylyk, Estimates for distribution of suprema of solutions to higherorder partial differential equations with random initial conditions, Modern Stoch. Theory Appl., vol. 7, iss. 1, pp.
–96, 2019.
Yu.V. Kozachenko, E.I. Ostrovskij, Banach spaces of random variables of sub-Gaussian type, Theory Probab. Math.
Stat., vol. 32, pp. 45–56, 1986.
X. Lan, Y. Xiao, Strong local nondeterminism of spherical fractional Brownian motion, Statist. Probab. Lett., vol. 135, pp. 44–50, 2018.
X. Lan, D. Marinucci, Y. Xiao, Strong local nondeterminism and exact modulus of continuity for spherical Gaussian fields, Stoch. Process. Appl., vol. 128, pp. 1294–1315, 2018.
A. Lang, C. Schwab, Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial
differential equations, Ann. Appl. Probab., vol. 25, no. 6, pp. 3047–3094, 2015.
D. Marinucci, G. Peccati, Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications, Cambridge University Press, 356 p., 2011.
D. Marinucci, S. Vadlamani, A note on global suprema of band-limited spherical random functions, Statist. Probab. Lett., vol. 96, pp. 141–148, 2015.
D. Marinucci, S. Vadlamani, High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere, Ann. Appl. Probab., vol. 26, no. 1, pp. 462–506, 2016.
O. Vasylyk, Strictly φ-sub-Gaussian quasi shot noise processes, Statistics Opt. Inform. Comput., vol. 5, no. 2, pp. 109–120, 2017.
M.I. Yadrenko, Spectral Theory of Random Fields, New York: Optimization Software, 259 p., 1983.
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