On the asymptotic behaviour of a simple growing point process model

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 existence of Cesaro limits for the sequences of respective moments for the displacement vectors), we obtain the limiting behaviour of the distribution of the point last added to the process and also that of the normalized mean measure of the point process as time goes to infinity.