Open Access
2013 Regular conditional distributions of continuous max-infinitely divisible random fields
Clément Dombry, Frédéric Eyi-Minko
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Electron. J. Probab. 18: 1-21 (2013). DOI: 10.1214/EJP.v18-1991


This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given observations $\{\eta(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is the point process representation of max-infinitely divisible processes by Giné, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. This paper offers new tools and perspective or prediction in extreme value theory together with numerous potential applications.


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Clément Dombry. Frédéric Eyi-Minko. "Regular conditional distributions of continuous max-infinitely divisible random fields." Electron. J. Probab. 18 1 - 21, 2013.


Accepted: 13 January 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1287.60066
MathSciNet: MR3024101
Digital Object Identifier: 10.1214/EJP.v18-1991

Primary: 60G70
Secondary: 60G25

Keywords: max-infinitely divisible process , Max-stable process , point process representation , regular conditional distribution

Vol.18 • 2013
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