We consider the following setting: Let $Z_{1},...,Z_{n}$ be iid random variables with distribution function $H_{Z}$ and $u>0$ a constant. We set $M:= \sup_{n\in N} \sum_{k=1}^{n} Z_{k} $.
In the Book Modelling Extremal Events for Insurance and Finance by Embrechts, Klüppelberg and Mikosch, Chapter 8.3.2 I found the following step:
$P(Z_{1}\leq x_{1},...,Z_{n}\leq x_{n},M>u) = \int_{-\infty}^{x_{1}} ...\int_{-\infty}^{x_{n}} P(M > u-y_{1}-...-y_{n}) dH_{Z}(y_{1})...dH_{Z}(y_{n})$
Why does this hold ? Is $M$ independent of the $Z_{k}$'s ?