Let $\varepsilon = \{\varepsilon_i\}_{i\in\mathbb{R}^d}$ be a collection of independent standard Poisson point processes on $\mathbb{R}^d$. That is, $\varepsilon_i$ are functions from a probability space to a space of simple counting measures on $\mathbb{R}^d$ (see e.g. definition 2.1.1 in https://inria.hal.science/hal-02460214/preview/PointProcesses45.pdf).
Let $\Phi$ be a random simple point process on $\mathbb{R}^d$. Denote $P_{\Phi}$ its distribution.
Does there exist a function $f$ such that $$f(\varepsilon) \overset{D}{=}P_{\Phi}\,\,\,\,\,\,\,\,?$$
I solved it for the case when $\Phi$ is Poisson point process with intensity function $\lambda_{\Phi}(x)$. Then, we can simply choose $f$ to be only function of $\varepsilon_1$ that transforms $\lambda_1$ (intensity of $\varepsilon_1$) to $\lambda_{\Phi}$. Since intensities are the same, by properties of Poisson point porcess, the entire distribution is the same. How can we proceed more generally?
