## Electronic Journal of Probability

### Regular conditional distributions of continuous max-infinitely divisible random fields

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 7, 21 pp.

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

https://projecteuclid.org/euclid.ejp/1465064232

Digital Object Identifier
doi:10.1214/EJP.v18-1991

Mathematical Reviews number (MathSciNet)
MR3024101

Zentralblatt MATH identifier
1287.60066

Subjects
Primary: 60G70: Extreme value theory; extremal processes

Rights

#### Citation

Dombry, Clément; Eyi-Minko, Frédéric. Regular conditional distributions of continuous max-infinitely divisible random fields. Electron. J. Probab. 18 (2013), paper no. 7, 21 pp. doi:10.1214/EJP.v18-1991. https://projecteuclid.org/euclid.ejp/1465064232

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