Independent marking theorem for Poisson point processes
= Independent marking theorem for Poisson point processes
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Independently assigning a mark to each point of a <Poisson point process>, with mark probabilities allowed to depend measurably on the point, produces independent Poisson point processes for the different marks. If the original intensity is $\lambda(x)\,dx$ and mark $j$ has probability $p_j(x)$, its intensity is $\lambda(x)p_j(x)\,dx$.