Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-15T06:30:39.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalent representations of max-stable processes via ℓp-norms

Part of: Stochastic processes

Published online by Cambridge University Press:  28 March 2018

Marco Oesting
Marco Oesting*
Affiliation:
Universität Siegen
*
* Postal address: Department Mathematik, Universität Siegen, Walter-Flex-Str. 3, 57072 Siegen, Germany. Email address: oesting@mathematik.uni-siegen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on ℓp-norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of an ℓp-norm-based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.

Keywords

Extreme value theoryReich–Shaby modelspectral representationstable tail dependence function

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G52: Stable processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 54 - 68
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. John Wiley, Chichester. CrossRefGoogle Scholar
[2]Berg, C., Christensen, J. P. R. and Ressel, P. (1984). Harmonic Analysis on Semigroups. Springer, New York. CrossRefGoogle Scholar
[3]De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204. Google Scholar
[4]Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown-Resnick random fields at a finite number of locations. Extremes 18, 301–314. CrossRefGoogle Scholar
[5]Dombry, C. and Eyi-Minko, F. (2012). Strong mixing properties of max-infinitely divisible random fields. Stoch. Process. Appl. 122, 3790–3811. CrossRefGoogle Scholar
[6]Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2015). Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. J. R. Statist. Soc. B 77, 239–265. CrossRefGoogle Scholar
[7]Engelke, S., Malinowski, A., Oesting, M. and Schlather, M. (2014). Statistical inference for max-stable processes by conditioning on extreme events. Adv. Appl. Prob. 46, 478–495. CrossRefGoogle Scholar
[8]Fougères, A.-L., Mercadier, C. and Nolan, J. P. (2013). Dense classes of multivariate extreme value distributions. J. Multivariate Anal. 116, 109–129. CrossRefGoogle Scholar
[9]Fougères, A.-L., Nolan, J. P. and Rootzén, H. (2009). Models for dependent extremes using stable mixtures. Scand. J. Statist. 36, 42–59. CrossRefGoogle Scholar
[10]Giné, E., Hahn, M. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165. CrossRefGoogle Scholar
[11]Gradshteyn, I. S. and Ryzhik, I. M. (1965). Tables of Integrals, Series, and Products. Academic Press, New York. Google Scholar
[12]Gumbel, É. J. (1960). Distributions des valeurs extrêmes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9, 171–173. Google Scholar
[13]Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd edn. Cambridge University Press. Google Scholar
[14]Kabluchko, Z. and Schlather, M. (2010). Ergodic properties of max-infinitely divisible processes. Stoch. Process. Appl. 120, 281–295. CrossRefGoogle Scholar
[15]Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Prob. 37, 2042–2065. CrossRefGoogle Scholar
[16]Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press. Google Scholar
[17]Molchanov, I. (2008). Convex geometry of max-stable distributions. Extremes 11, 235–259. CrossRefGoogle Scholar
[18]Oesting, M., Kabluchko, Z. and Schlather, M. (2012). Simulation of Brown-Resnick processes. Extremes 15, 89–107. CrossRefGoogle Scholar
[19]Oesting, M., Schlather, M. and Zhou, C. (2018). Exact and fast simulation of max-stable processes on a compact set using the normalized spectral representation. Bernoulli 24, 1497–1530. CrossRefGoogle Scholar
[20]Penrose, M. D. (1992). Semi-min-stable processes. Ann. Prob. 20, 1450–1463. CrossRefGoogle Scholar
[21]Reich, B. J. and Shaby, B. A. (2012). A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Statist. 6, 1430–1451. CrossRefGoogle ScholarPubMed
[22]Reich, B. J., Shaby, B. A. and Cooley, D. (2014). A hierarchical model for serially-dependent extremes: a study of heat waves in the western US. J. Agric. Biol. Environ. Statist. 19, 119–135. CrossRefGoogle Scholar
[23]Ressel, P. (2013). Homogeneous distributions—and a spectral representation of classical mean values and stable tail dependence functions. J. Multivariate Anal. 117, 246–256. CrossRefGoogle Scholar
[24]Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York. Google Scholar
[25]Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5, 33–44. CrossRefGoogle Scholar
[26]Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90, 139–156. CrossRefGoogle Scholar
[27]Sebille, Q., Fougères, A.-L. and Mercadier, C. (2017). Modeling extreme rainfall: a comparative study of spatial extreme value models. Spat. Statist. 21, 187–208. CrossRefGoogle Scholar
[28]Shaby, B. A. and Reich, B. J. (2012). Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland. Environmetrics 23, 638–648. CrossRefGoogle Scholar
[29]Smith, R. L. (1990). Max–Stable Processes and Spatial Extremes. Unpublished manuscript. Google Scholar
[30]Stephenson, A. G., Shaby, B. A., Reich, B. J. and Sullivan, A. L. (2015). Estimating spatially varying severity thresholds of a forest fire danger rating system using max-stable extreme-event modeling. J. Appl. Meteor. Climatol. 54, 395–407. CrossRefGoogle Scholar