Large Deviations of Poisson Cluster Processes

Poisson cluster processes are one of the most important classes of point process models (see Daley and Vere-Jones[7] and Møller; Waagepetersen[24]). They are natural models for the location of objects in the space, and are widely used in point process studies whether theoretical or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or Hawkes processes (Hawkes[12,13]; Hawkes and Oakes[14]). From a theoretical point of view, Hawkes processes combine both a Poisson cluster process representation and a simple stochastic intensity representation. Poisson cluster processes found applications in cosmology, ecology and epidemiology; see, respectively, Neyman and Scott[25], Brix and Chadoeuf[5], and Møller[19,20]. Hawkes processes are particularly appealing for seismological applications. Indeed, they are widely used as statistical models for the standard activity of earthquake series; see the papers by