  We consider a sequence $(\xi_n)_{n\ge1}$ of $i.i.d.$ random values living in
the domain of attraction of an extreme value distribution. For such sequence,
there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for every $n\ge
1$, such that the sequence $(X_n)$ defined by
$X_n=(\max(\xi_1,...,\xi_n)-b_n)/a_n$ converges in distribution to a non
degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as
an Euler scheme with decreasing step of an ergodic Markov process solution to a
SDE with jumps and we derive a functional limit theorem for the sequence
$(X_n)$ from some methods used in the long time numerical approximation of
ergodic SDE's.
