极值理论与SDE的长时间逼近的联系
A connection between extreme value theory and long time approximation of SDE's
摘要：
我们考虑一个序列$(ξ_n)_{n\ge1}$的$i.i.d.$随机值生活在域的吸引力的极值分布。对于这样的序列，存在$(a_n)$和$(b_n)$，对于每个$n\ge 1$具有$a_n>0$和$b_n\in\ER$，使得由$X_n=(\max(ξ_1,...,ξ_n)-b_n)/a_n$定义的序列$(X_n)$在分布上收敛到非退化分布。本文证明了$(X_n)$可以看作是遍历马氏过程解的一个步长递减的Euler格式，并利用遍历马氏过程长时间数值逼近的一些方法，得到了序列$(X_n)$的一个泛函极限定理.
英文摘要：
We consider a sequence $(ξ_n)_{n\ge1}$ 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 $X_n=(\max(ξ_1,...,ξ_n)-b_n)/a_n$ 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.
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