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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
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Abstract

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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 

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