How can I prove, that a Poisson Point Process $\mathcal{P}_n$ on $\mathbb{R}^d$ of the region $B_{r_n}(x)$, $x\in\mathbb{R}^d$ with intensity $nf$ is stochastically dominated by a Poisson random variable with mean $ r_*f^* $. $f^*$ is the supremum of the probability density and $r_*= \sup_{n\geq 1} \omega_d nr_n^d < \infty $, where $\omega_d$ is the volume of a $d$-dimensional unit ball.
In short:
$$\mathcal{P}_n(B_{r_n}(x)) \leq \operatorname{Poi}(r_*f^*).$$
I know that for Stochastic Dominance I have to compare the Cumulative distribution functions. My first question would be: What is the CDF of the given point process?
Thanks in advance!