Take a Poisson process with some mean (say $\lambda t = \mu$) and compound it with a Binomial distribution. Meaning every time a Poisson arrival hits, instead of getting one event, we generate a Binomial random variable with parameters $l$ and $p$ and this tells us the number of events.
Alternately, consider a deterministically compounded Poisson process where every time we have an arrival, we get a deterministic number of events.
I had a conjecture that with a Compound Poisson, with Binomial compounding, as we increase the $l$ parameter of the Binomial (number of tosses), it would start looking more and more like a deterministically compounded Poisson process with $[lp]$ as the deterministic compounding factor. So, I made a QQ plot for these two distributions for various values of $l$ (see below). The parameters were: (mean of Poisson process: $119$, Binomial $l$: varies in the plot, Binomial $p$: $.333$). I observe the following:
- Indeed, as we increase $l$, the QQ plot seems to move towards a straight line.
- This doesn't seem to be a completely clean pattern though, with some values of $l$ intermittently faring worse than another (further from a straight line) even though they are larger.
What could be the reasons for these observations?
The reason I suspected this pattern was that a Binomial random variable has a variance of $lp-lp^2$ and a mean of $lp$. As we increase the value of $l$, the difference between the mean and variance should start increasing (variance getting smaller and smaller than the mean). So, the process should start looking more and more like a deterministically compounded process. However, this is certainly not a concrete argument.