Categorical thinning of a Poisson claim count ($N_i\text{ independent},N_i\sim \mathrm{Pois}(\lambda p_i)$)

Assign each of $N\sim \mathrm{Pois}(\lambda )$ claims with independent category labels to one of finitely many categories with probabilities $p_i$. Conditional counts have a multinomial distribution with probability generating function $({\sum }_i{p}_i{z}_i{)}^N$. Averaging over $N$ gives $\exp (\lambda ({\sum }_i{p}_i{z}_i-1))={\prod }_i\exp (\lambda p_i(z_i-1))$. This proves independent category counts and supports discrete-severity compound Poisson distribution representations.