I want to derive the theoretical relation between the parameters in a point process model for extremes and the parameters in the GPD model for extremes. I'm following Coles - An introduction to Statistical Modeling of Extreme Values, p.131-132.
My attempt so far: Let $Y_i$ be independent measurements of data, and define the sequence of point processes.
$$ N_n = \left\{ (i/(n+1),Y_i) \, : \, i=1, \dots,n \right\}. $$
Now, for sufficiently large u, on regions of the form $(0,1)\times [u, \infty),$ $N_n$ is approximately a poisson process with intensity measure on $A_z=[t_1,t_2] \times (z,\infty)$ given by
$$ \Lambda(A_z)=(t_2-t_1) \left[ 1+ \gamma \left( \frac{z-\mu}{\sigma} \right) \right]^{-1/\gamma}. $$
The idea here is to calculate $$ \mathbb{P}(Y_i \leq z \; | \; Y_i >u) $$ and compare it to the general form of the GPD distribution, namely $$ H(z;\hat{\sigma},\hat{\gamma}) = 1 - \left[ 1 + \hat{\gamma} \left( \frac{z}{\hat{\sigma}} \right) \right]^{-1/\hat{\gamma}}.$$ So, for $z>u$, $$ \mathbb{P}(Y_i > z \; | \; Y_i >u) = \frac{\mathbb{P}(Y_i > z \; , \; Y_i >u)}{\mathbb{P}( Y_i >u)} = \frac{\mathbb{P}(Y_i > z)}{\mathbb{P}( Y_i >u)} = *$$
My problem: Unfortunately here I'm stuck. In Coles book they factorize the intensity as
$$ \Lambda(A_z) = \Lambda_1([t_1,t_2])\times \Lambda_2([z,\infty)), $$
where
$$ \Lambda_1([t_1,t_2]) = (t_2-t_1) \quad \text{ and } \Lambda_2([z,\infty)) = \left[ 1+ \gamma \left( \frac{z-\mu}{\sigma} \right) \right]^{-1/\gamma} $$
and then simply write
$$ * = \frac{\Lambda_2([z,\infty))}{\Lambda_2([u,\infty))} = \frac{n^{-1}[1+\gamma(z-\mu)/\sigma]^{-1/\gamma}}{n^{-1}[1+\gamma(u-\mu)/\sigma]^{-1/\gamma}}.$$
Can anyone explain this step to me? And where does the $n^{-1}$ come from? This is very unclear to me.
I think I might have figured it out, so I'll anwer for completeness.
$$N_n(A_z) = \text{number of points in the set } A_z \in \operatorname{Po}(\Lambda(A_z)).$$ Let $$ (i-1)/(n+1) < t_1 < i/(n+1) < t_2 < (i+1)/(n+1). $$
Then $$ N(A_z) = {\bf1}( Y_i > z ). $$
Now, $$ \mathbb{P}(Y_i > z) = \mathbb{E}[{\bf1}( Y_i > z )] = \mathbb{E}[N(A_z)] = \Lambda(A_z). $$
This explains the $*$-step above. From there it is simple to rewrite the expression as
$$ \mathbb{P}(Y_i > z \; | \; Y_i > u) = \left[ 1 + \gamma \left( \frac{z-u}{\sigma + \gamma(u- \mu)}\right) \right]^{-1/\gamma} \quad \Rightarrow $$ $$ Y_i -u \; | \; Y_i > u \in H(z;\sigma + \gamma(u-\mu),\gamma). $$
This implies the desired relation between the parameters.