We present several results refining and extending those of M.F.Neuts and A.S. Alfa ("Pair formation in a MAP with two event labels", J. Appl. Probab. 41 (2004), to appear) weak convergence of the pair formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair formation process to a Poisson one actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuples formation).

We consider a finite simple point process in a finite-dimensional Euclidean space 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.
 

Convenor:Aidan Sudbury