under smaller and smaller thresholds, for which we make use of classical approximations due to Balkema and de Haan (1974[@bal]) which stands as follows, firstly in the commonly known setting of exceedances over increasing thresholds. Denote $\widetilde{R}:=1/R$.

\[Thm de Haan\]For $\widetilde{R}$ of distribution $F$ belonging to the maximum domain of attraction of an extreme value distribution with tail index $c$, i.e. $F\in MDA(c)$, it holds that: There exists $a=a(s)>0$, such that: $$\lim\limits_{s \rightarrow \infty}\sup_{0\le x< \infty} 
\left\lvert 
\frac{1 - F\left(x + s\right)}{1 - F\left( s \right)} -  \left(1 - G_{(c,a}(x)\right)
\right\rvert = 0$$ where $G_{(c,a)}$ is defined through $~$$$G_{(c,a)}(x)=1-\exp \left\{ -\int_{0}^{\frac{x}{a}}\left[ (1+ct)_{+}\right]
^{-1}dt\right\}$$ where $a>0$ and $c\in \mathbb{R}$.

The distribution $G$ is the Generalized Pareto distribution $GPD(c,a)$ is defined explicitly through$$1-G(x)=\left\{ 
\begin{array}{l}
(1+\frac{c}{a}x)^{-1/c}\text{ when }c\neq 0 \\ 
\exp (-\frac{x}{a})\text{ when }c=0\end{array}\right.$$where $x\geq 0$ for $c\geq 0$ and $0\leq x\leq -\frac{a}{c}$ if $c<0.$

Generalized Pareto distributions enjoy invariance through threshold conditioning, an important property for our sake. Indeed it holds, for $\widetilde{R}\sim GDP(c,a)$ and $x>s$,

$$\mathbb{P}\left( \widetilde{R}>x\mid \widetilde{R}>s\right) =\left( 1+\frac{c(x-s)}{a+cs}\right) ^{-1/c}  \label{r}$$

We therefore state:

\[Prop stability GPD\]When $\widetilde{R}\sim GPD(c,a)$ then, given $\left( \widetilde{R}>s\right) $, the r.v. $\widetilde{R}-s$ follows a $GPD(c,a+cs)$.

The GPD’s are on the one hand stable under thresholding 