Max stable processes

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Max-stable processes extend the multi-dimensional extreme value theory to the infinite-dimensional case. Similar to the one- and multi-dimensional case, such a process occurs as a limit value of the maxima of appropriately normalized independent copies of a stochastic process.

definition

Let be any index set. A stochastic process is called max-stable if there are normalization constants such that it holds for independent copies of the process ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle a_{n}(t)>0,b_{n}(t)\in \mathbb {R} }$ ${\displaystyle X_{1},\dotsc ,X_{n}}$ ${\displaystyle X}$

${\displaystyle \left({\frac {\max _{i=1}^{n}X_{i}(t)-b_{n}(t)}{a_{n}(t)}},t\in T\right){\overset {d}{=}}(X(t),t\in T)}$.

The one-dimensional marginal distributions of a max-stable process are given by one of the three univariate extreme value distributions. In the case of Fréchet distributed margins d. H. the normalization constants can be chosen as follows: . ${\displaystyle \mathbb {P} (X(t)\leq x)=e^{-1/x}}$ ${\displaystyle a_{n}=n,b_{n}=0}$

General

Be independent copies of the stochastic process . There are now normalization constants , so apply for and and the process is not degenerate, it is a max-stable process. A max-stable process with simple Fréchet-distributed margins can be constructed using its spectral representation. ${\displaystyle Y_{i},i=1,\dotsc ,n}$ ${\displaystyle Y}$ ${\displaystyle c_{n}(t)>0,d_{n}(t)\in \mathbb {R} }$ ${\displaystyle \max _{i=1}^{n}{\frac {Y_{i}(t)-d_{n}(t)}{c_{n}(t)}}\rightarrow X(t)}$ ${\displaystyle n\to \infty }$ ${\displaystyle t\in T}$ ${\displaystyle X}$ ${\displaystyle X}$

Web links

• Matthias Löwe, Extreme Value Theory, Lecture Notes Uni Münster, 2008

Individual evidence

1. ^ Maximilian Zott: Extreme Value Theory in Higher Dimensions - Max-Stable Processes and Multivariate Records . ( uni-wuerzburg.de [accessed October 7, 2019]).
2. ↑ Laurens de Haan: A Spectral Representation for Max-stable Processes. In: The Annals of Probability. 1984.