= Constant intensity does not imply independence
The same <intensity measure> can belong to very different <point processes>. A <Poisson point process> with a locally finite <intensity measure> $\mu$ adds two requirements: each finite-intensity window count has the <Poisson distribution> of mean $\mu(W)$, and counts in pairwise <disjoint observation windows> are mutually <independent>. These requirements determine its <probability law>; they do not follow from the formula for $\mu$.<Lectures on the Poisson Process by Günter Last and others>[Section 3.1, Definition 3.1 and Proposition 3.2, printed pp. 19-20: Poisson count laws, independence, and uniqueness]{ref}
A <randomly shifted lattice point process>, a unit-rate <Poisson point process>, and the <shared random rate gives dependent point counts> construction all satisfy $\mu(A)=|A|$ on the real line. Nevertheless, their counts in $[0,1)$ have respectively <variance> zero, one, and two. Counts in the <disjoint observation windows> $[0,1/2)$ and $[1/2,1)$ have respectively <covariance> $-1/4$, zero, and $1/4$. These exact calculations show why measuring mean density cannot identify a spacing or <statistical dependence> model.
\Image[/point-process-equal-intensity-different-count-laws.png]
{title=Equal intensity permits regular spacing, Poisson counts, and shared-rate dependence}
{description=All three laws have <intensity measure> $\mu(A)=|A|$. Left: illustrative realizations of a <randomly shifted lattice point process>, a unit-rate <Poisson point process>, and the <shared random rate gives dependent point counts> construction conditional on rate two; the latter also has a completely empty outcome with <probability> $1/2$. Right: exact <probability mass functions> of the count in $[0,1)$, with the axis displayed through seven. Equal <expected values> of one coexist with <variances> zero, one, and two. Realizations illustrate spacing; the bars describe laws, not frequencies estimated from those realizations.}
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