Regular conditional distributions of max infinitely divisible processes

TL;DR

This paper derives an explicit formula for the conditional distribution of max-stable processes given observations, enhancing prediction methods in extreme value theory through a detailed analysis of the underlying point process structure.

Contribution

It introduces a novel explicit expression for the regular conditional distribution of max-infinitely divisible processes, extending previous results and providing new tools for prediction in extreme value analysis.

Findings

Explicit conditional distribution formula as a mixture over hitting scenarios

Extension of previous results to more general max-stable processes

Potential applications in extreme value prediction and analysis

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 Gin\'e, 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 related result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. We believe this work offers new tools and perspective for prediction in extreme value theory together with numerous potential applications.