On the asymptotic behaviour of a simple growing point process model

Abstract

We consider a finite simple point process in space Rd evolving in discrete time in the following way. Starting with an arbitrary initial configuration, at each time step a point is chosen at random from the process according to a certain distribution, and then k new points are added to the process at locations, each obtained by adding an independent random vector to the location of the chosen "mother" point. The k "displacement vectors" are independent of each other and of the past evolution of the process, and follow a given common distribution that can depend on the time step (while the value of k remains fixed over time). Under mild moment conditions (uniform integrability and the existen..