Central limit theorem for independent, non-identically distributed, finite discrete random variables I am positive, that there exists a version of CLT stating that, the distribution of the sum of infinite many independent, nonidentical, finite discrete random variables, is Gaussian. I just could not find the reference any more right now. Does anyone have the resource?
Edit: all the distributions are nondegenerate (in my case), but as the discussion and answer below pointed out, as long as the sum of their variances goes infinity, CLT applies.

I understand the most general CLT theorem to-date invokes the Lindeberg Feller condition to prove the asymptotic limiting normal distribution holds.
That is if a sequence of independent random variables $X_k$ satisfies the Lindeberg condition:
$$\lim_{n \rightarrow \infty} \frac{1}{s_n^2} \sum_{i=1}^n \mathbb{E} \left[ (X_k - \mu_k) ^2 \cdot \mathcal{I}(|X_k - \mu_k| > \epsilon s_n)\right] = 0 \quad \forall \epsilon > 0$$
then the usual statement of the CLT applies. Note we make no statement about the distribution of $X_k$. The Lindeberg condition can be thought of as requiring that deviations get "swallowed" up with the standard error of the mean. It's interesting to consider some counterexamples to the Lindeberg condition, particularly where a finite mean exists by the WLLN.
