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Suppose there is a homogeneous Poisson point process (PPP) $ \phi $ in the plane $\mathbb R^2$.

Can we prove that the expected area in the plan that is closer to a given point $x_i \in \phi$ than to any other point in $\phi$, is finite?

i.e., $$E\{y:\mid\mid x-x_i\mid\mid \leq \mid\mid x-x_j\mid\mid j\in \phi, j\neq i\}<\infty ?$$

It seems very intuitive but I would like to understand how to prove it.

Thanks

triomphe
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1 Answers1

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Some point $x$ at distance $r$ from $x_i$ is in the domain whose mean area $A$ you want to compute if and only if the disk of center $x$ and radius $r$ contains no other point of the Poisson process. This happens with probability $\mathrm e^{-\lambda\pi r^2}$ hence $$ A=\int_0^\infty\mathrm e^{-\lambda\pi r^2}2\pi r\mathrm dr=\frac1\lambda. $$ In particular, the mean area $A$ is finite.

Did
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