# Neigbourhood of Poisson point process

Let $C$ be a unit circle and $P$ be a Poisson point process on $C$ with intensity $1$. For each point $p$ of the process we draw an interval $(p-\epsilon, p+\epsilon)$ for some $\epsilon>0$. Further let's denote by $L$ Lebesgue's measure of the union of the intervals.

What is the law of $L$?

What can be done easily is for example calculating $\mathbb{E}L$. Namely, we have

$$L=\int_{C} 1_{(x-\epsilon, x+\epsilon)\cap P\neq \emptyset}\text{d}x,$$ which gives $$\mathbb{E}L=\int_{C} \mathbb{P}((x-\epsilon, x+\epsilon)\cap P\neq \emptyset)\text{d}x=(2\pi)\mathbb{P}((-\epsilon, +\epsilon)\cap P\neq \emptyset) = (2\pi)(1-\exp(-2\epsilon)).$$

In principle this method can be use to calculate recursively moments of arbitrary order but it seems very cumbersome.

• I think an exact formula that works for general $\epsilon > 0$ is out of reach since even the formula for the moments of $L$ will depend on the value of $\epsilon$. For instance, the formula for $E[L^k]$ will be different when $\epsilon > \frac{\pi}{k}$ and when $\epsilon < \frac{\pi}{k}$ since if $\epsilon > \frac{\pi}{k}$ then it is possible for $k$ intervals of width $2\epsilon$ to cover the entire circle. – Jon Peterson Jan 10 '14 at 14:49
• Have you tried the Laplace functional of L? – Arash Jan 10 '14 at 16:10
• The Laplace transform was my first attempt. But I failed :). – Piotr Miłoś Jan 11 '14 at 10:19