On pair and tuple formation under independent Poisson or renewal arrival processes

Abstract

We present several results refining and extending those of Neuts and Alfa on 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 process 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-tuple formation).