A compound Poisson process involves events arriving according to a Poisson process with rate $\lambda$. However, with each arrival, instead of there being just one event, we generate a binomial random variable (parameters $l$ and $p$) and that dictates the number of events. I'm trying to work out the PMF of such a distribution. Let $M(t)$ be this compound point process and $N(t)$ the underlying Poisson process. We then get:
$$P(M(t) = j) = \sum\limits_{m=0}^\infty P(M(t)=j|N(t)=m)P(N(t)=m)$$
Conditional on $N(t)=m$, we basically end up summing $m$ binomial random variables with the same $p$ parameter. This becomes another binomial with parameters $lm$ and $p$. So we get:
$$P(M(t)=j)=\sum\limits_{m=0}^\infty {lm \choose j}p^j(1-p)^{lm-j} \left(\frac{e^{-\lambda t} (\lambda t)^m}{m!}\right)$$
I've been thinking about how to proceed with this summation all day, but can't make progress.
Also, I have good reason to beleive that for large $l$,
$$ P(M(t)=j) \tilde{=} P(N(t)=[j l p])$$
Is it obvious to anyone that this might be the case?