and on the other appear as the limit distribution for thresholding operations. This chain of arguments is quite usual in statistics, motivating the recourse to the ubiquous normal or stable laws for additive models. This plays in favor of GPD’s as modelling the distribution of $\widetilde{R}$ for excess probability inference. Due to the lack of memory property, the exponential distribution which appears as a possible limit distribution for excess probabilities in Theorem \[Thm de Haan\] do not qualify for modelling. Moreover since we handle variables $R$ which can approach $0$ arbitrarily (i.e. unbounded $\widetilde{R}$) the parameter $c$ is assumed positive.

Turning to the context of the minimal admissible constraint, we make use of the r.v. $R=1/\widetilde{R}$ and proceed to the corresponding change of variable.

When $c>0$, the distribution function of the r.v. $R$ writes for nonnegative $x$: $$\label{GPD 1}
F_{c,a}(x)=(1+\frac{c}{ax})^{-1/c}.$$

For $0<x<u$, the conditional distribution of $R$ given $\left\{ R<u\right\} $ is $$\mathbb{P}(R<x\mid R<u)=\left( 1-\frac{c(\frac{1}{x}-\frac{1}{u})}{a+\frac{c}{u}}\right) ^{-1/c}$$ which proves that the distribution of $R$ is stable under threshold conditioning with parameter $\left( a_{u},c\right) $ with $$a_{u}=a+\frac{c}{u}.  \label{transition param a}$$In practice at each step $j$ in the procedure the stress level $s_{j}$ equals the corresponding threshold $1/\widetilde{s}_{j}$ , a right quantile of 