Inspired by real-world claim arrivals, the model balances interesting stylized facts (such as dependence across the components, over-dispersion and the clustering of claims) with a high level of mathematical tractability (including estimation, sampling and convergence results for large portfolios) and can thus be applied in various contexts (such as risk management and pricing of (re-)insurance contracts). The authors provide a detailed analysis of the proposed probabilistic model, discussing its relation to the existing literature, its statistical properties, different estimation strategies as well as possible applications and extensions.