Let $ X_1,X_2,...,X_N $ be independent Poisson variables with different parameters $\lambda_n$ that potentially can be quite small (e.g., $10^{-4}$). I am interested in the distribution of $Y$ which is the linear combination
$Y=\sum_{n=1}^N a_n X_n $,
with the $a_n \in \mathbb{R}$, and $N$ is quite large ($N>>1000$), but finite.
Although I think this could be a standard textbook problem, I was googling quite a bit and did not find a solution. It intuitively makes sense that the central limit theorem (CLT) can be used here, because the Poisson distribution can be approximated by a normal distribution with the mean equaling the variance, and then we are left with a large sum of (scaled) normal distributions which itself is normal. The parameters $\lambda_n$ are very small so the normal approximation will be quite poor, but that is probably taken care of by the large number of summands. So my question is: is there a reference to a textbook with a formal proof that the distribution of $Y$ is normal, or can someone write down a formal proof here? I tried to get started with a proof using the Lindeberg CLT, but end up with a sum for which I don't know how to come up with an inequality that shows convergence.