Hostname: page-component-77f85d65b8-v2srd Total loading time: 0 Render date: 2026-04-17T20:50:16.722Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure and representations of max-stable processes

Part of: Stochastic processes Ergodic theory

Published online by Cambridge University Press:  01 July 2016

Yizao Wang  and
Stilian A. Stoev
Yizao Wang*
Affiliation:
University of Michigan
Stilian A. Stoev*
Affiliation:
University of Michigan
*
∗ Postal address: Department of Statistics, The University of Michigan, 439 W. Hall, 1085 S. University, Ann Arbor, MI 48109-1107, USA.
∗ Postal address: Department of Statistics, The University of Michigan, 439 W. Hall, 1085 S. University, Ann Arbor, MI 48109-1107, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max-stable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

Keywords

Max-stable processclassificationmax-linear isometryspectral representationnonsingular flowco-spectral function

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G10: Stationary processes 60G52: Stable processes 37A50: Relations with probability theory and stochastic processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 855 - 877
Copyright
Copyright © Applied Probability Trust 2010 

References

Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Balkema, A. A. and Resnick, S. I. (1977). Max-infinite divisibility. J. Appl. Prob. 14, 309–319.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Prob. 14, 732–739.Google Scholar
Cohn, D. L. (1972). Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrscheinlichkeitsth. 22, 161–165.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1993). Prediction of stationary max-stable processes. Ann. Appl. Prob. 3, 497–525.Google Scholar
De Haan, L. (1978). A characterization of multidimensional extreme-value distributions. Sankhyā A 40, 85–88.Google Scholar
De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.Google Scholar
De Haan, L. and Pickands, J. III (1986). Stationary min-stable stochastic processes. Prob. Theory Relat. Fields 72, 477–492.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165.Google Scholar
Hardin, C. D. Jr. (1981). Isometries on subspaces of L p . Indiana Univ. Math. J. 30, 449–465.Google Scholar
Hardin, C. D. Jr. (1982). On the spectral representation of symmetric stable processes. J. Multivariate Anal. 12, 385–401.Google Scholar
Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes 12, 401–424.Google Scholar
Kabluchko, Z. and Schlather, M. (2010). Ergodic properties of max-infinitely divisible processes. Stoch. Process. Appl. 120, 281–295.Google Scholar
Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Prob. 37, 2042–2065.Google Scholar
Krengel, U. (1969). Darstellungssätze für Strömungen und Halbströmungen. II. Math. Ann. 182, 1–39.CrossRefGoogle Scholar
Krengel, U. (1985). Ergodic Theorems. Walter de Gruyter, Berlin.Google Scholar
Oodaira, H. (1972). On Strassen's version of the law of the iterated logarithm for Gaussian processes. Z. Wahrscheinlichkeitsth. 21, 289–299.CrossRefGoogle Scholar
Pipiras, V. (2007). Nonminimal sets, their projections and integral representations of stable processes. Stoch. Process. Appl. 117, 1285–1302.Google Scholar
Pipiras, V. and Taqqu, M. S. (2002). The structure of self-similar stable mixed moving averages. Ann. Prob. 30, 898–932.CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2004). Stable stationary processes related to cyclic flows. Ann. Prob. 32, 2222–2260.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. and Roy, R. (1991). Random usc functions, max-stable processes and continuous choice. Ann. Appl. Prob. 1, 267–292.Google Scholar
Rosiński, J. (1994). On uniqueness of the spectral representation of stable processes. J. Theoret. Prob. 7, 615–634.Google Scholar
Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187.Google Scholar
Rosiński, J. (2000). Decomposition of stationary α-stable random fields. Ann. Prob. 28, 1797–1813.CrossRefGoogle Scholar
Rosiński, J. (2006). Minimal integral representations of stable processes. Prob. Math. Statist. 26, 121–142.Google Scholar
Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2, 365–377.Google Scholar
Samorodnitsky, G. (2005). Null flows, positive flows and the structure of stationary symmetric stable processes. Ann. Prob. 33, 1782–1803.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sikorski, R. (1964). Boolean Algebras, 2nd edn. Academic Press, New York.Google Scholar
Stoev, S. A. (2008). On the ergodicity and mixing of max-stable processes. Stoch. Process. Appl. 118, 1679–1705.Google Scholar
Stoev, S. A. and Taqqu, M. S. (2005). Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes. Extremes 8, 237–266.CrossRefGoogle Scholar
Wang, Y. and Stoev, S. A. (2009). On the structure and representations of max-stable processes. Tech. Rep. 487, Department of Statistics, University of Michigan. Available at http://arxiv.org/abs/0903.3594.Google Scholar