It is well known that the superposition of $N$ Poisson processes is itself a Poisson process with an intensity given by $\sum_{n=1}^{N} \lambda _{n}$.
Conversely a superposition including any non-Poisson component processes is not Poisson.
However, the superposition of sparse processes converges to a Poisson process as $N\rightarrow \infty $ and as the sparseness increases. This is referred to on Wikipedia as the Palm-Khintchine theorem: a recent paper with discussion of the original work is given by Schuhmacher 2005.
If I have a collection of sampled processes, the superposition of which is well modeled as a Poisson process, under what conditions of $N$ & $\hat{\lambda_{n}}$ can I conclude that all component processes are Poisson?
P.S. This problem is known and has been studied in queuing theory. Newell 1984 provides a condition that if the component arrival processes satisfy $n(1-\rho)^2 {\gg } 1$, where $\rho$ is the utilisation, then the queue will approximate a M/M/1 queue (Poisson arrival process). How does this formula apply to the more general problem outlined above?
