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

We consider a sequence ( ξ n ) 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 ( a n ) and ( b n ) , with a n > 0 and b n ∈ R for every n ≥ 1 , such that the sequence ( X n ) defined by X n = ( max ( ξ 1 , … , ξ n ) − b n ) / a n converges in distribution to a non-degenerated distribution. s paper, we show that ( X n ) 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 ( X n ) from some methods used in the long time numerical approximation of ergodic SDEs.