Define a Poisson random measure. State and prove the Product Theorem for the jump times $J_{n}$ of a Poisson process with constant rate $\lambda$ and independent random variables $Y_{n}$ with law $\mu$. Write down the corresponding result for a Poisson process $\Pi$ in a space $E=\mathbb{R}^{d}$ with rate $\lambda(x)(x \in E)$ when we associate with each $X \in \Pi$ an independent random variable $m_{X}$ with density $\rho(X, d m)$.

Prove Campbell's Theorem, i.e. show that if $M$ is a Poisson random measure on the space $E$ with intensity measure $\nu$ and $a: E \rightarrow \mathbb{R}$ is a bounded measurable function then

$$\mathbf{E}\left[e^{\theta \Sigma}\right]=\exp \left(\int_{E}\left(e^{\theta a(y)}-1\right) \nu(d y)\right)$$

where

$$\Sigma=\int_{E} a(y) M(d y)=\sum_{X \in \Pi} a(X)$$

Stars are scattered over three-dimensional space $\mathbb{R}^{3}$ in a Poisson process $\Pi$ with density $\nu(X)\left(X \in \mathbb{R}^{3}\right)$. Masses of the stars are independent random variables; the mass $m_{X}$ of a star at $X$ has the density $\rho(X, d m)$. The gravitational potential at the origin is given by

$$F=\sum_{X \in \Pi} \frac{G m_{X}}{|X|}$$

where $G$ is a constant. Find the moment generating function $\mathbf{E}\left[e^{\theta F}\right]$.

A galaxy occupies a sphere of radius $R$ centred at the origin. The density of stars is $\nu(\mathbf{x})=1 /|\mathbf{x}|$ for points $\mathbf{x}$ inside the sphere; the mass of each star has the exponential distribution with mean $M$. Calculate the expected potential due to the galaxy at the origin. Let $C$ be a positive constant. Find the distribution of the distance from the origin to the nearest star whose contribution to the potential $F$ is at least $C$.