Electronic Journal of Probability

Regular conditional distributions of continuous max-infinitely divisible random fields

Clément Dombry and Frédéric Eyi-Minko

Full-text: Open access

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

Permanent link to this document
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
Secondary: 60G25: Prediction theory [See also 62M20]

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

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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