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Extrema of a Gaussian random field: Berman’s sojourn time method. (English) Zbl 1513.60054

Summary: In this paper we devote ourselves to extending Berman’s sojourn time method, which is thoroughly described in [S. M. Berman, Ann. Probab. 2, 999–1026 (1974; Zbl 0298.60026); Ann. Probab. 10, 1–46 (1982; Zbl 0498.60035); Sojourns and extremes of stochastic processes. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Book & Software (1992; Zbl 0809.60046)], to investigate the tail asymptotics of the extrema of a Gaussian random field over \([0, T]^d\) with \(T \in (0, \infty)\).

MSC:

60G60 Random fields
60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes
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References:

[1] Berman, S. M., Sojourns and extremes of Gaussian processes, The Annals of Probability, 2, 6, 999-1026 (1974) · Zbl 0298.60026
[2] Berman, S. M., Sojourns and extremes of stationary processes, The Annals of Probability, 10, 1, 1-46 (1982) · Zbl 0498.60035
[3] Berman, S. M., Sojourns and Extremes of Stochastic Processes (1992), Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA · Zbl 0809.60046
[4] Pickands, J. III, Maxima of stationary Gaussian processes, Z Wahrscheinlichkeitstheorie und Verw Gebiete, 7, 2, 190-223 (1967) · Zbl 0158.16702
[5] Pickands, J. III, Asymptotic properties of the maximum in a stationary Gaussian process, Transactions of the American Mathematical Society, 145, 75-86 (1969) · Zbl 0206.18901
[6] Pickands, J. III, Upcrossing probabilities for stationary Gaussian processes, Transactions of the American Mathematical Society, 145, 51-73 (1969) · Zbl 0206.18802
[7] Piterbarg, V. I., On the paper by J. Pickands “Upcrossing probabilities for stationary Gaussian processes”, Vestnik Moskovskogo Universiteta Serija I. Matematika, Mehanika, 27, 5, 25-30 (1972) · Zbl 0241.60031
[8] Piterbarg, V. I.; Prisjažnjuk, V. P., Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process, Theory of Probability & Mathematical Statistics, 18, 2, 131-144 (1979) · Zbl 0433.60030
[9] Piterbarg, V. I., Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0841.60024
[10] Hüsler, J.; Piterbarg, V. I., Extremes of a certain class of Gaussian processes, Stochastic Processes and Their Applications, 83, 2, 257-271 (1999) · Zbl 0997.60057
[11] Piterbarg, V. I., Large deviations of a storage process with fractional Brownian motion as input, Extremes, 4, 2, 147-164 (2001) · Zbl 1003.60053
[12] Dȩbicki, K.; Hashorva, E.; Soja-Kukieła, N., Extremes of homogeneous Gaussian random fields, Journal of Applied Probability, 52, 1, 55-67 (2015) · Zbl 1315.60059
[13] Dȩbicki, K.; Hashorva, E.; Liu, P., Extremes of γ-reflected Gaussian process with stationary increments, ESAIM Probability & Statistics, 21, 495-535 (2017) · Zbl 1393.60034
[14] Hashorva, E.; Ji, L., Piterbarg theorems for chi-processes with trend, Extremes, 18, 1, 37-64 (2015) · Zbl 1315.60042
[15] Dȩbicki, K.; Hashorva, E.; Ji, L.; Tabiś, K., Extremes of vector-valued Gaussian processes: Exact asymptotics, Stochastic Processes and Their Applications, 125, 11, 4039-4065 (2015) · Zbl 1321.60108
[16] Dȩbicki, K.; Engelke, S.; Hashorva, E., Generalized Pickands constants and stationary max-stable processes, Extremes, 20, 493-517 (2017) · Zbl 1379.60041
[17] Dȩbicki, K.; Hashorva, E.; Liu, P., Extremes of Gaussian random fields with regularly varying dependence structure, Extremes, 20, 333-393 (2017) · Zbl 1373.60069
[18] Dȩbicki, K.; Hashorva, E., Approximation of supremum of max-stable stationary processes & Pickands constants, Journal of Theoretical Probability, 33, 444-464 (2020) · Zbl 1462.60039
[19] Dȩbicki, K.; Michna, Z.; Peng, X. F., Approximation of sojourn times of Gaussian processes, Methodology and Computing in Applied Probability, 21, 4, 1183-1213 (2019) · Zbl 1447.60063
[20] Dȩbicki, K.; Hashorva, E.; Liu, P., Uniform tail approximation of homogenous functionals of Gaussian fields, Advances in Applied Probability, 49, 4, 1037-1066 (2017) · Zbl 1425.60036
[21] Berman, S. M., Sojourns and extremes of a diffusion process on a fixed interval, Advances in Applied Probability, 14, 4, 811-832 (1982) · Zbl 0494.60076
[22] Berman, S. M., Sojourns of stationary processes in rare sets, The Annals of Probability, 11, 4, 847-866 (1983) · Zbl 0562.60043
[23] Berman, S. M., A sojourn limit theorem for Gaussian processes with increasing variance, Stochastics, 13, 4, 281-298 (2007) · Zbl 0559.60038
[24] Berman, S. M., Extreme sojourns for random walks and birth-and-death processes. Communications in Statistics, Stochastic Models, 2, 3, 393-408 (1986) · Zbl 0611.60061
[25] Berman, S. M., Sojourns and extremes of a stochastic process defined as a random linear combination of arbitrary functions. Communications in Statistics, Stochastic Models 4, 49, 1, 1-43 (1988) · Zbl 0642.60026
[26] Berman, S. M., Extreme sojourns of diffusion processes, Annals of Probability, 16, 1, 361-374 (1988) · Zbl 0637.60089
[27] Berman, S. M., Sojourn times in a cone for a class of vector Gaussian processes, SIAM Journal on Applied Mathematics, 49, 2, 608-616 (1989) · Zbl 0671.60032
[28] Berman, S. M., Sojourns of vector Gaussian processes inside and outside spheres, Probability Theory and Related Fields, 66, 4, 529-542 (1984) · Zbl 0561.60042
[29] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge: Cambridge University Press, Cambridge · Zbl 0617.26001
[30] Fernique, X., Regularité des trajectoires des fonctions aléatoires Gaussiennes (in French), Lecture Notes in Math, 480, 1-96 (1975) · Zbl 0331.60025
[31] Adler, R. J., An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes (1990), Hayward, Californla: Institute of Mathematical Statistics, Hayward, Californla · Zbl 0747.60039
[32] Adler, R. J.; Taylor, J. E., Random Fields and Geometry (2007), New York: Springer, New York · Zbl 1149.60003
[33] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1998), New York: Springer, New York
[34] Cheng, S. H., The Fundementals of Measurement Theory and Probability Theory (in Chinese) (2004), Beijing: Peking University Press, Beijing
[35] Li, X. P., Fundementals of Probability Theory (in Chinese) (2010), Beijing: Higher Education Press, Beijing
[36] Dȩbicki, K.; Hashorva, E.; Ji, L. P., Parisian ruin over a finite-time horizon, Science China Mathematics, 59, 3, 557-572 (2016) · Zbl 1341.60024
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