Mapping theorem for Poisson point processes

Let $\Pi$ be a Poisson point process with intensity measure $\mu$ and let $f$ be measurable. If the pushforwardis locally finite, then the image counting measure is Poisson with intensity $f_{\ast }\mu$. If one requires a simple point process, the pushforward must also be diffuse, so distinct source points do not produce coincident image points with positive probability.

For a homogeneous Poisson point process of intensity $\lambda$ in ${\mathbb{R}}^d$, mapping $x$ to $v_d|x{|}^d$, where $v_d$ is the unit-ball volume, gives a rate-$\lambda$ Poisson process on $[0,\infty )$. Mapping to $|x{|}^d$ instead gives rate $\lambda v_d$.

For intensity $\lambda$ in ${\mathbb{R}}^d$, let $R_k$ be the distance to the $k$th closest point and put $\rho =\lambda v_d$. Then $R_k^d$ is the $k$th arrival time of a rate-$\rho$ Poisson process, so