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Let $M$ be a Poisson cloud with intensity $dxdy$ over $\mathbb{R}^2$. We denote $M_{\theta}$ and $M_R$ as the Poisson clouds on $R/2π \mathbb{Z}$ and $\mathbb{R}^+$, respectively, obtained from M by considering the angles and distances from the origin of the points in M. Provide the intensities of these two clouds. Are they independent?

I do not know how to compute the intensity of those two clouds. Do I have to integrate over $[0, 2\pi]$ for the the intensity of $M_{\theta}$ and over $[0,R]$ for $M_R$.

How do I know they are independent ?

Thanks.

user1240705
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  • Does it make sense to consider an infinite number of points on the compact space $R/2\pi Z?$ – Gérard Letac Jan 09 '24 at 12:01
  • Thanks. I do not understand your question. Why $\mathbb{R}/ 2 \pi \mathbb{Z}$ cannot have an infinite number of points ? How this can help me to find the intensity of $M_{\theta}$ ? – user1240705 Jan 09 '24 at 14:05
  • I guess the term "Poisson cloud" is not ubiquitous in the literature. Is it simply a 2-dimensional Poisson point process (PPP)? If so, I am not sure if $M_{\theta}$ defines a PPP on $\mathbb{R}/2\pi\mathbb{Z}$ because $M_{\theta}$ is almost surely dense in $\mathbb{R}/2\pi\mathbb{Z}$ (whereas any "sensible" PPP on $\mathbb{R}/2\pi\mathbb{Z}$ must be discrete, hence finite). – Sangchul Lee Jan 09 '24 at 16:30

1 Answers1

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A Poisson cloud on a measured space $(X,\mu) $ is such that the number $N(A)$ of points in $A\subset X$ is Poisson with mean $\mu(A).$ If $X$ is compact the measure $\mu$ is bounded and $N(X)$ is finite. I do not see what is the candidate for $\mu$ in your case since the image of the Lebesgue measure $dxdy$ by the map $$(x,y)\mapsto \frac{(x,y)}{\sqrt{x^2+y^2}} $$ does not exist.

Gérard Letac
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