Implicit max-stable extremal integrals
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Implicit max-stable extremal integrals D. Kremer1 Received: 19 August 2019 / Revised: 2 July 2020 / Accepted: 17 July 2020 / © The Author(s) 2020
Abstract Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function f ≥ 0. From an application point of view, one is rather interested in extreme loss events that occur relative to f than in the corresponding extreme values itself. In this context, so-called f -implicit α-Fr´echet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in Goldbach (2016) by developing a stochastic integral of a deterministic function g ≥ 0 with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see Stoev and Taqqu Extremes 8, 237–266 (2005)) and, at the same time, reveals striking parallels. Keywords Implicit max-stable distributions · Independently scattered random sup-measures · Stochastic integrals · Implicit max-stable processes AMS 2000 Subject Classifications 60G57 · 60G60 · 60G70
1 Introduction The theory of implicit extreme values is highly motivated by application, such as hydrology (see the introductory example in Scheffler and Stoev (2017)), and tries to analyze the circumstances in which several impact factors lead to extreme loss or damage. Hence, different from classical extreme value theory (shortly: EVT), this perspective is less interested in the attained extreme values than in the study of complex systems that cause these extreme values. Particularly, the isolated impacts (components) of the system do not have to be extreme in any sense, but can still contribute to such extreme loss events.
D. Kremer
[email protected] 1
Department Mathematik, Universit¨at Siegen, 57068 Siegen, Germany
D. Kremer
In this context it is reasonable to assume that the connection between the impact factors and the related loss is known. More precisely, throughout the paper we consider a fixed function f : Rd → [0, ∞) that serves as some kind of loss function, depending on d ≥ 1 impact factors. For technical reasons, we have to assume that f fulfills the following properties, which still appear natural for most examples: (i) (ii) (iii)
f is continuous. f (x) = 0 if and only if x = 0. f is 1-homogeneous, i.e. we have f (λx) = λf (x) for every λ ≥ 0 and x ∈ R.
Turning over to probability, we consider a random vector X = (X(1) , ..., X(d) ) modeling the joint behavior of the d impact factors. Then, for a sequence (Xj )j ∈N of identically distributed and independent (i.i.d.) random vectors, a major subject of classical multivariate EVT is to understand the asymptotic behavior (under possible normalization) of k Mk := Xj := max Xj (1.1) j =1
j =1,...,k
as k → ∞, where the maximum is meant component-wise. Sometimes, one is also interested in the study of maxj =1,...k f (Xj ), which leads to an associated univariate problem. In con
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