Articles

EXTREMA OF A GAUSSIAN RANDOM FIELD: BERMAN’S SOJOURN TIME METHOD

  • Liwen CHEN ,
  • Xiaofan PENG
Expand
  • School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

Received date: 2021-03-10

  Revised date: 2022-05-13

  Online published: 2022-11-02

Supported by

The second author was partially supported by National Natural Science Foundation of China (11701070, 71871046) and Ronglian Scholarship Fund.

Abstract

In this paper we devote ourselves to extending Berman’s sojourn time method, which is thoroughly described in [1–3], to investigate the tail asymptotics of the extrema of a Gaussian random field over [0,T]d with T ∈ (0, ∞).

Cite this article

Liwen CHEN , Xiaofan PENG . EXTREMA OF A GAUSSIAN RANDOM FIELD: BERMAN’S SOJOURN TIME METHOD[J]. Acta mathematica scientia, Series B, 2022 , 42(5) : 1831 -1842 . DOI: 10.1007/s10473-022-0508-y

References

[1] Berman S M. Sojourns and extremes of Gaussian processes. The Annals of Probability. 1974, 2(6): 999–1026
[2] Berman S M. Sojourns and extremes of stationary processes. The Annals of Probability. 1982, 10(1): 1–46
[3] Berman S M. Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole Statisitics/Probability Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992
[4] Pickands III J. Maxima of stationary Gaussian processes. Z Wahrscheinlichkeitstheorie und Verw Gebiete, 1967, 7(2): 190–223
[5] Pickands III J. Asymptotic properties of the maximum in a stationary Gaussian process. Transactions of the American Mathematical Society, 1969, 145: 75–86
[6] Pickands III J. Upcrossing probabilities for stationary Gaussian processes. Transactions of the American Mathematical Society, 1969, 145: 51–73
[7] Piterbarg V I. On the paper by J. Pickands “Upcrossing probabilities for stationary Gaussian processes”. Vestnik Moskovskogo Universiteta Serija I. Matematika, Mehanika, 1972, 27(5): 25–30
[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, 1979, 18(2): 131–144
[9] Piterbarg V I. Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. Providence, RI: American Mathematical Society, 1996
[10] Hüsler J, Piterbarg V I. Extremes of a certain class of Gaussian processes. Stochastic Processes and Their Applications, 1999, 83(2): 257–271
[11] Piterbarg V I. Large deviations of a storage process with fractional Brownian motion as input. Extremes, 2001, 4(2): 147–164
[12] Dȩbicki K, Hashorva E, Soja-Kukieła N. Extremes of homogeneous Gaussian random fields. Journal of Applied Probability, 2015, 52(1): 55–67
[13] Dȩbicki K, Hashorva E, Liu P. Extremes of γ-reflected Gaussian process with stationary increments. ESAIM Probability & Statistics, 2017, 21: 495–535
[14] Hashorva E, Ji L. Piterbarg theorems for chi-processes with trend. Extremes, 2015, 18(1): 37–64
[15] Dȩbicki K, Hashorva E, Ji L, Tabiś K. Extremes of vector-valued Gaussian processes: Exact asymptotics. Stochastic Processes and Their Applications, 2015, 125(11): 4039–4065
[16] Dȩbicki K, Engelke S, Hashorva E. Generalized Pickands constants and stationary max-stable processes. Extremes, 2017, 20: 493–517
[17] Dȩbicki K, Hashorva E, Liu P. Extremes of Gaussian random fields with regularly varying dependence structure. Extremes, 2017, 20: 333–393
[18] Dȩbicki K, Hashorva E. Approximation of supremum of max-stable stationary processes & Pickands constants. Journal of Theoretical Probability, 2020, 33: 444–464
[19] Dȩbicki K, Michna Z, Peng X F. Approximation of sojourn times of Gaussian processes. Methodology and Computing in Applied Probability, 2019, 21(4): 1183–1213
[20] Dȩbicki K, Hashorva E, Liu P. Uniform tail approximation of homogenous functionals of Gaussian fields. Advances in Applied Probability, 2017, 49(04): 1037–1066
[21] Berman S M. Sojourns and extremes of a diffusion process on a fixed interval. Advances in Applied Probability, 1982, 14(4): 811–832
[22] Berman S M. Sojourns of stationary processes in rare sets. The Annals of Probability. 1983, 11(4): 847–866
[23] Berman S M. A sojourn limit theorem for Gaussian processes with increasing variance. Stochastics, 2007, 13(4): 281–298
[24] Berman S M. Extreme sojourns for random walks and birth-and-death processes. Communications in Statistics. Stochastic Models, 1986, 2(3): 393–408
[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, 1988, 49(1): 1–43
[26] Berman S M. Extreme sojourns of diffusion processes. Annals of Probability, 1988, 16(1): 361–374
[27] Berman S M. Sojourn times in a cone for a class of vector Gaussian processes. SIAM Journal on Applied Mathematics, 1989, 49(2): 608–616
[28] Berman S M. Sojourns of vector Gaussian processes inside and outside spheres. Probability Theory and Related Fields, 1984, 66(4): 529–542
[29] Bingham N H, Goldie C M, Teugels J L. Regular Variation. Cambridge: Cambridge University Press, 1987
[30] Fernique X. Regularité des trajectoires des fonctions aléatoires Gaussiennes (in French). Lecture Notes in Math, 1975, 480: 1–96
[31] Adler R J. An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Lecture Notes-Monograph Series, 12. Hayward, Californla: Institute of Mathematical Statistics, 1990
[32] Adler R J, Taylor J E. Random Fields and Geometry. New York: Springer, 2007
[33] Karatzas I, Shreve S E. Brownian Motion and Stochastic Calculus. 2nd ed. New York: Springer, 1998
[34] Cheng S H. The Fundementals of Measurement Theory and Probability Theory (in Chinese). Beijing: Peking University Press, 2004
[35] Li X P. Fundementals of Probability Theory (in Chinese). Beijing: Higher Education Press, 2010
[36] Dȩbicki K, Hashorva E, Ji L P. Parisian ruin over a finite-time horizon. Science China Mathematics, 2016, 59(3): 557–572
Options
Outlines

/