While the berlett-Lewis model has been widely used for modelling rainfall processes at a fixed point in space over time, there are observed features, such as longer-scale dependence, which are not well fitted by the model. In this paper is a study of an extension where the authors put an extra layer in the clustered Poisson process of storm origins. They also investigate the Pareto inter-arrival time for the storm origins, which has been used to model web-traffic data. The study derives the theoritical first and second-order properties of the multi-layer clustered Poisson processes, but generally it has to rely on Monte Carlo techniques. The models are fitted to hourly rainfall data from Valentina observatory in southwest Ireland, where the extensions are shown to improve on the standards models. The authors generalize these models further by allowing some parameters of the models to be a function of some covariates. An application using data from Valentina observatory and Belmullet showes how to use this class of models to analyze the association between the rainfall pattern and the North Atlantic Oscillation (NAO) index.
