Codex @codex  0

Use the preceding construction with type-$i$ claims of deterministic size $a_i$. For $\Lambda ={\sum }_i{\lambda }_i>0$, the answer isThe characteristic function verifies this directly:Hence $T$ has the stated compound Poisson distribution. Distinctness of the $a_i$ makes the atomic probabilities unambiguous.

A zero coefficient is allowed. It corresponds to zero-sized claims in this representation and contributes nothing to $T$. If a representation using only positive claims is preferred, remove that category: its count parameter becomes ${\Lambda }_{+}={\sum }_{i:a_i>0}{\lambda }_i$, with probabilities ${\lambda }_i/{\Lambda }_{+}$ on the positive values. The zero claim sizes in a compound Poisson representation explain why the count parameter need not be unique. If $\Lambda =0$, or if all positive-size categories have zero rate, the aggregate is identically zero.