×

zbMATH — the first resource for mathematics

Stationary max-stable fields associated to negative definite functions. (English) Zbl 1208.60051
Summary: Let \(W_i\), \(i\in\mathbb N\), be independent copies of a zero-mean Gaussian process \(\{W(t)\), \(t\in\mathbb R^d\}\) with stationary increments and variance \(\sigma^2(t)\). Independently of \(W_i\), let \(\sum^\infty_{i=1}\delta_{U_i}\) be a Poisson point process on the real line with intensity \(e^{-y}dy\). We show that the law of the random family of functions \(\{V_i(\cdot)\), \(i\in\mathbb N\}\), where \(V_i(t)=U_i+W_i(t)-\sigma^2(t)/2\), is translation invariant. In particular, the process \(\eta(t)=\bigvee^\infty_{i=1}V_i(t)\) is a stationary max-stable process with standard Gumbel margins. The process \(\eta\) arises as a limit of a suitably normalized and rescaled pointwise maximum of \(n\) i.i.d. stationary Gaussian processes as \(n\to\infty\) if and only if \(W\) is a (nonisotropic) fractional Brownian motion on \(\mathbb R^d\). Under suitable conditions on \(W\), the process \(\eta\) has a mixed moving maxima representation.

MSC:
60G70 Extreme value theory; extremal stochastic processes
60G15 Gaussian processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aaronson, J. and Denker, M. (1998). Characteristic functions of random variables attracted to 1-stable laws. Ann. Probab. 26 399-415. · Zbl 0937.60005
[2] Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups . Springer, New York. · Zbl 0308.31001
[3] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003
[4] Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab. 14 732-739. JSTOR: · Zbl 0384.60055
[5] Brown, M. (1970). A property of Poisson processes and its application to macroscopic equilibrium of particle systems. Ann. Math. Statist. 41 1935-1941. · Zbl 0233.60057
[6] Buishand, T., de Haan, L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem. Ann. Appl. Stat. 2 624-642. · Zbl 1273.62258
[7] Davis, R. A. and Resnick, S. I. (1989). Basic properties and prediction of max-ARMA processes. Adv. in Appl. Probab. 21 781-803. JSTOR: · Zbl 0716.62098
[8] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194-1204. · Zbl 0597.60050
[9] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory : An Introduction . Springer, New York. · Zbl 1101.62002
[10] de Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in C [0, 1]. Ann. Probab. 29 467-483. · Zbl 1010.62016
[11] de Haan, L. and Pereira, T. T. (2006). Spatial extremes: Models for the stationary case. Ann. Statist. 34 146-168. · Zbl 1104.60021
[12] de Haan, L. and Pickands, J. (1986). Stationary min-stable stochastic processes. Probab. Theory Related Fields 72 477-492. · Zbl 0577.60034
[13] de Haan, L. and Zhou, C. (2008). On extreme value analysis of a spatial process. REVSTAT 6 71-81. · Zbl 1153.62074
[14] Deheuvels, P. (1983). Point processes and multivariate extreme values. J. Multivariate Anal. 13 257-272. · Zbl 0519.60045
[15] Eddy, W. F. and Gale, J. D. (1981). The convex hull of a spherically symmetric sample. Adv. in Appl. Probab. 13 751-763. JSTOR: · Zbl 0477.60031
[16] Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Related Fields 87 139-165. · Zbl 0688.60031
[17] Hooghiemstra, G. and Hüsler, J. (1996). A note on maxima of bivariate random vectors. Statist. Probab. Lett. 31 1-6. · Zbl 0879.60056
[18] Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283-286. · Zbl 0679.62038
[19] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3 . Oxford Univ. Press, New York. · Zbl 0771.60001
[20] Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhyā Ser. A 32 369-378. · Zbl 0218.60039
[21] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes . Springer, New York. · Zbl 0518.60021
[22] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces : Isoperimetry and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete 3 23 . Springer, Berlin. · Zbl 0748.60004
[23] Marcus, M. B. (1972). Upper bounds for the asymptotic maxima of continuous Gaussian processes. Ann. Math. Statist. 43 522-533. · Zbl 0241.60032
[24] Pickands, J. III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51-73. · Zbl 0206.18802
[25] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148 . Amer. Math. Soc., Providence, RI. · Zbl 0841.60024
[26] Resnick, S. I. (1987). Extreme Values , Regular Variation , and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4 . Springer, New York. · Zbl 0633.60001
[27] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33-44. · Zbl 1035.60054
[28] Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90 139-156. · Zbl 1035.62045
[29] Stoev, S. A. and Taqqu, M. S. (2005). Extremal stochastic integrals: A parallel between max-stable processes and \alpha -stable processes. Extremes 8 237-266. · Zbl 1142.60355
[30] Stoev, S. A. (2008). On the ergodicity and mixing of max-stable processes. Stochastic Process. Appl. 118 1679-1705. · Zbl 1184.60013
[31] Zhang, Z. and Smith, R. L. (2004). The behavior of multivariate maxima of moving maxima processes. J. Appl. Probab. 41 1113-1123. · Zbl 1122.60052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.