Skip to main content Accessibility help

Space‒time max-stable models with spectral separability

  • Paul Embrechts (a1), Erwan Koch (a1) and Christian Robert (a2)


Natural disasters may have considerable impact on society as well as on the (re-)insurance industry. Max-stable processes are ideally suited for the modelling of the spatial extent of such extreme events, but it is often assumed that there is no temporal dependence. Only a few papers have introduced spatiotemporal max-stable models, extending the Smith, Schlather and Brown‒Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role to space and the temporal dynamics are not explicit. In order to overcome these defects, we introduce spatiotemporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous- and discrete-time versions. We then consider particular Markovian cases with a max-autoregressive representation and discuss their properties. Finally, we briefly propose an inference methodology which is tested through a simulation study.


Corresponding author

Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address:
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address:
ISFA, Université Lyon 1, 50 Avenue Tony Garnier, 69366 Lyon Cedex 07, France. Email address:


Hide All
Beirlant, J.,Goegebeur, Y.,Segers, J. and Teugels, J. (2004).Statistics of Extremes: Theory and Applications.John Wiley,New York.
Bienvenüe, A. and Robert, C. Y. (2014).Likelihood based inference for high-dimensional extreme value distributions. Preprint. Available at
Buhl, S. and Klüppelberg, C. (2015).Anisotropic Brown-Resnick space-time processes: estimation and model assessment. Preprint. Available at
Coles, S. (2001).An Introduction to Statistical Modeling of Extreme Values.Springer,London.
Cooley, D.,Naveau, P. and Poncet, P. (2006).Variograms for spatial max-stable random fields. In Dependence in Probability and Statistics (Lecture Notes Statist. 187), eds P. Bertail, P. Doukhan and P. Soulier,Springer,New York, pp. 373390.
Davis, R. A. and Resnick, S. I. (1989).Basic properties and prediction of max-ARMA processes.Adv. Appl. Prob. 21,781803.
Davis, R. A.,Klüppelberg, C. and Steinkohl, C. (2013a).Max-stable processes for modeling extremes observed in space and time.J. Korean Statist. Soc. 42,399414.
Davis, R. A.,Klüppelberg, C. and Steinkohl, C. (2013b).Statistical inference for max-stable processes in space and time.J. R. Statist. Soc. B 75,791819.
Dombry, C. and Eyi-Minko, F. (2014).Stationary max-stable processes with the Markov property.Stoch. Process. Appl. 124,22662279.
Embrechts, P.,Klüppelberg, C. and Mikosch, T. (1997).Modelling Extremal Events for Insurance and Finance.Springer,Berlin.
Haan, L. de (1984).A spectral representation for max-stable processes.Ann. Prob. 12,11941204.
Haan, L. de and Ferreira, A. (2006).Extreme Value Theory: An Introduction.Springer,New York.
Haan, L. de and Pickands, J. (1986).Stationary min-stable stochastic processes.Prob. Theory Relat. Fields 72,477492.
Hairer, M. (2010).Convergence of Markov processes. Probability at Warwick course. Available at
Huser, R. and Davison, A. (2014).Space-time modelling of extreme events.J. R. Statist. Soc. B 76,439461.
Kabluchko, Z. and Schlather, M. (2010).Ergodic properties of max-infinitely divisible processes.Stoch. Process. Appl. 120,281295.
Kabluchko, Z.,Schlather, M. and Haan, L. de (2009).Stationary max-stable fields associated to negative definite functions.Ann. Prob. 37,20422065.
Mardia, K. V. and Jupp, P. E. (1999).Directional Statistics.John Wiley,New York.
Meinguet, T. (2012).Maxima of moving maxima of continuous functions.Extremes 15,267297.
Naveau, P.,Guillou, A.,Cooley, D. and Diebolt, J. (2009).Modelling pairwise dependence of maxima in space.Biometrika 96,117.
Padoan, S. A.,Ribatet, M. and Sisson, S. A. (2010).Likelihood-based inference for max-stable processes.J. Amer. Statist. Assoc. 105,263277.
Penrose, M. D. (1992).Semi-min-stable processes.Ann. Prob. 20,14501463.
Resnick, S. I. (1987).Extreme Values, Regular Variation, and Point Processes.Springer,New York.
Ribatet, M. (2015).SpatialExtremes. R Package, version 2.0–2. Available at
Schlather, M. (2002).Models for stationary max-stable random fields.Extremes 5,3344.
Schlather, M. and Tawn, J. A. (2003).A dependence measure for multivariate and spatial extreme values: properties and inference.Biometrika 90,139156.
Smith, R. L. (1990).Max-stable processes and spatial extremes. Unpublished manuscript.
Strokorb, K.,Ballani, F. and Schlather, M. (2015).Tail correlation functions of max-stable processes: construction principles, recovery and diversity of some mixing max-stable processes with identical TCF.Extremes 18,241271.
Swiss Re., (2014).Natural catastrophes and man-made disasters in 2013: large losses from floods and hail; Haiyan hits the Philippines.Sigma 1/2014,Swiss Re, Zürich.


MSC classification

Space‒time max-stable models with spectral separability

  • Paul Embrechts (a1), Erwan Koch (a1) and Christian Robert (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed