= Poisson offspring clusters directed by their parent process
{c}
{title2=$\Lambda_\Pi(A)=\mu\sum_{z\in\Pi}\kappa(A-z)$}
When each parent has a Poisson number of independently displaced offspring, the conditional offspring process has the displayed directing measure. For disjoint target sets, <multinomial distribution> splitting of a parent's Poisson count gives joint <probability generating function> $\exp(\mu\sum_j\kappa(A_j-z)(s_j-1))$. Multiply over <independent> parents to obtain <independent> Poisson counts conditional on the parents. Compactly supported displacement ensures local finiteness for a locally finite parent process.
// Destination: discrete-probability-distribution.bigb
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