Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 27L (Applied Probability)

Let $\Pi$ be a Poisson point process on ${\mathbb{R}}^d$ with intensity measurewhere $\lambda$ is nonnegative and locally integrable. Let $f:{\mathbb{R}}^d\to {\mathbb{R}}^s$ be measurable and suppose its pushforward measureis locally finite. The mapping theorem for Poisson point processes says that the image counting measureis a Poisson random measure with intensity $\nu$.

For $f(\Pi )$ to be a spatial Poisson process in the usual simple sense, one also assumes that $\nu$ is diffuse:This prevents collisions with positive probability. If $\nu$ is absolutely continuous, its Radon--Nikodym derivative $\tilde{\lambda }$ is the image intensity function, characterized by

Write $r=(x^2+y^2+z^2{)}^{1/2}$, so $f(x,y,z)=r^3$. For $t\ge 0$,whose volume isThus the pushforward intensity of $[0,t]$ isThe pushforward is locally finite and diffuse, so the radial volume transform of a homogeneous Poisson point process and the mapping theorem show that

Put $T_k=R_k^3$. Part (i) identifies $T_k$ as the $k$th arrival time of a ratePoisson process. Hence $T_k$ has the gamma densityThe change of variables $t=r^3$, with $dt=3r^{2}dr$, gives the kth-nearest-neighbour distance in a homogeneous Poisson point process: