Poisson offspring clusters directed by their parent process (${\Lambda }_{\Pi }(A)=\mu {\sum }_{z\in \Pi }\kappa (A-z)$)

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