Definition given:
A stochastic process $\{X(t),t\geq 0\}$ is a compound Poisson process (used for modelling aggregate claim size) if:
$$X(t) = \sum_{i=1}^{N(t)} Y_i$$
for $t\geq 0$ and where
$\{N(t),t\geq 0\}$ is a Poisson process (number of claims/frequency of claims)
$\{Y_i,i=1,2,\ldots,n\}$ are iid random variables (not necessarily Poisson) and are the claim sizes.
Since $X(t)$ will follow Poisson distribution, why does adding non Poisson iid r.v.s $Y_i$ give a Poisson random variable? Does it have something to do with the fact that the number of claims are Poisson distributed?
$X(t)$ need not be a Poisson process. For example, if $Y_i\equiv 2$, then $X(t)=2N(t)$ is not a Poisson process.