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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.

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  • $\begingroup$ 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. $\endgroup$
    – Jon Peterson
    Jan 10, 2014 at 14:49
  • $\begingroup$ Have you tried the Laplace functional of L? $\endgroup$
    – Arash
    Jan 10, 2014 at 16:10
  • $\begingroup$ The Laplace transform was my first attempt. But I failed :). $\endgroup$
    – Piotr Miłoś
    Jan 11, 2014 at 10:19

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