Abstract
We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0, 1] are characterized by their “distribution functions” G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of “sojourn time transformation” and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W = 1 + log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes.
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References
Aulbach, S., Bayer, V., Falk, M.: A multivariate piecing-together approach with an application to operational loss data. Bernoulli 18(2), 455–475 (2012). doi:10.3150/10-BEJ343
Balkema, A.A., de Haan, L.: Residual life time at great age. Ann. Probab. 2(5), 792–804 (1974). doi:10.1214/aop/1176996548
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics. Wiley, New York (1999)
Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2(2), 624–642 (2008). doi:10.1214/08-AOAS159
Falk, M., Hüsler, J., Reiss, R.D.: Laws of Small Numbers: Extremes and Rare Events, 3rd edn. Birkhäuser, Basel (2010)
Ferreira, A., de Haan, L.: The generalized Pareto process; with application. Tech. rep. (2012). arXiv:1203.2551v1
Giné, E., Hahn, M., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Relat. Fields 87(2), 139–165 (1990). doi:10.1007/BF01198427
de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194–1204 (1984). doi:10.1214/aop/1176993148
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006). See http://people.few.eur.nl/ldehaan/EVTbook.correction.pdf and http://home.isa.utl.pt/~anafh/corrections.pdf for corrections and extensions
de Haan, L., Lin, T.: On convergence toward an extreme value distribution in C[0, 1]. Ann. Probab. 29(1), 467–483 (2001). doi:10.1214/aop/1008956340
de Haan, L., Pickands III, J.: Stationary min-stable stochastic processes. Probab. Theory Relat. Fields 72(4), 477–492 (1986). doi:10.1007/BF00344716
Hofmann, M.: On the hitting probability of max-stable processes. Tech. rep., University of Würzburg (2012, submitted). arXiv:1206.5913v1
Molchanov, I.: Theory of Random Sets. Probability and Its Applications. Springer, London (2005)
Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975). doi:10.1214/aos/1176343003
Rootzén, H., Tajvidi, N.: Multivariate generalized Pareto distributions. Bernoulli 12(5), 917–930 (2006). doi:10.3150/bj/1161614952
Takahashi, R.: Characterizations of a multivariate extreme value distribution. Adv. Appl. Probab. 20(1), 235–236 (1988). doi:10.2307/1427279
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Aulbach, S., Falk, M. & Hofmann, M. On max-stable processes and the functional D-norm. Extremes 16, 255–283 (2013). https://doi.org/10.1007/s10687-012-0160-3
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DOI: https://doi.org/10.1007/s10687-012-0160-3
Keywords
- Max-stable process
- Functional D-norm
- Functional domain of attraction
- Copula process
- Generalized Pareto process
- Takahashi’s theorem
- Sojourn times transformation