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On convergence toward an extreme value distribution in $C\left[0,1\right]$. (English) Zbl 1010.62016

From the paper: The structure of extreme value distributions in infinite-dimensional space is well known. We characterize the domain of attraction of such extreme-value distributions in the framework of E. Giné, M.G. Hahn and P. Vatan [Probab. Theory Relat. Fields 87, No. 2, 139-165 (1990; Zbl 0688.60031)]. We intend to use the result for statistical applications.

The two northern provinces, Friesland and Groningen, of the Netherlands are almost completely below sea level. Since there are no natural coast defenses like sand dunes, the area is protected against inundation by a long dike. Since there is no subdivision of the area by dikes, a break in the dike at any place could lead to flooding of the entire area. This leads to the following mathematical problem.

Suppose we have a deterministic function $f$ defined on $\left[0,1\right]$ (representing the top of the dike). Suppose we have i.i.d. random functions ${\xi }_{1},{\xi }_{2},\cdots$ defined on $\left[0,1\right]$ (representing observations of high tide water levels monitored along the coast). The question is: how can we estimate

$P\left\{{\xi }_{i}\left(t\right)\le f\left(t\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}i=1,\cdots ,k,0\le t\le 1\right\}=P\left\{\underset{1\le i\le k}{max}{\xi }_{i}\left(t\right)\le f\left(t\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le t\le 1\right\}$

on the basis of $n$ observed independent realizations of the process $\xi$ $\left(n$ large)? Now a typical feature of this kind of problems is that none of the observed processes $\xi$ come even close to the boundary $f$ that is, during the observation period there has not been any flooding-damage. This means that we have to extrapolate the distribution of $\xi$ far into the tail. Since nonparametric methods cannot be used, we resort to a limit theory; that is we imagine that $n\to \infty$ but in doing so we wish to keep the essential feature that the observations are far from the boundary. This leads to the assumption that $f$ is not a fixed function when $n\to \infty$ but that in fact $f$ depends on $n$ and moves to the upper boundary of the distribution of $\xi$ when $n\to \infty$. Another way of expressing this is that we assume that the left-hand side in the second inequality has a limit distribution after normalization. So in order to answer this question, we need a limit theory for the pointwise maximum of i.i.d. random functions and this is the subject of the present paper.

##### MSC:
 62E20 Asymptotic distribution theory in statistics 60G70 Extreme value theory; extremal processes (probability theory) 62G32 Statistics of extreme values; tail inference 62F05 Asymptotic properties of parametric tests 62F99 Parametric inference
##### Keywords:
extreme values; convergence in C[0,1]