A connection between extreme value theory and long time approximation of SDEs

We consider a sequence (ξn)n≥1(ξn)n≥1 of i.i.d.   random values residing in the domain of attraction of an extreme value distribution. For such a sequence, there exist (an)(an) and (bn)(bn), with an>0an>0 and bn∈Rbn∈R for every n≥1n≥1, such that the sequence (Xn)(Xn) defined by Xn=(max(ξ1,…,ξn)−bn)/anXn=(max(ξ1,…,ξn)−bn)/an converges in distribution to a non-degenerated distribution.In this paper, we show that (Xn)(Xn) can be viewed as an Euler scheme with a decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence (Xn)(Xn) from some methods used in the long time numerical approximation of ergodic SDEs.