# Central Limit Theorem with Bounded Sum of Variances?

I have a sequence of bounded independent random variables $$X_1,...,X_n,...$$ satisfying $$\sum_{i=1}^{\infty} \mathbb{E}[X_i] < \infty$$, $$\sum_{i=1}^{\infty} Var[X_i] < \infty$$. Most versions of the central limit theorem I've found online require $$\sum_{i=1}^{\infty} Var[X_i] = +\infty$$ in order to be applied. Is there some limit theorem which can give me the limiting distribution of $$Pr[\frac{\sum_{i=1}^{N}{X_i}-\mathbb{E}[X_i]}{\sqrt{\sum_{i=1}^{N} Var[X_i]}}\leq y].$$

Is there a reason why the central limit theorem would not apply?

Heuristically, if the sum of the variances isn't infinite, there is some residual shape information in the sum about some of the individual variables. For example, if $$\mathrm{var}[X_1]=\epsilon\sum_i \mathrm{var}[X_i]$$ then $$X_1$$ makes up $$\epsilon>0$$ of the limiting random variable and the shape of $$X_i$$ (tails, moments, etc) has $$\epsilon$$ effect on the shape of the limiting distribution.
One way to think about why there's a problem is the Levy-Cramér theorem, which says that if $$Y_1$$ and $$Y_2$$ are independent and not constant, and $$Y_1+Y_2$$ has a Normal distribution, then both $$Y_1$$ and $$Y_2$$ have Normal distributions.
Now take $$Y_1$$ to be $$X_1$$ and $$Y_2$$ to be the sum of the rest of the sequence. If $$\sum_i \mathrm{var}[X_i]=S^2<\infty$$, then $$Y_1$$ and $$Y_2$$ are non-constant independent random variables. Unless they are both Normal, their sum isn't Normal -- and so the sum of the series isn't Normal. You can see this argument breaks down if $$S^2$$ is infinite, as then $$Y_1/S$$ is constant.
In special cases you can think about this in terms of moments. For example, suppose $$Y_2$$ is Normal but $$Y_1$$ has non-zero skewness. Then $$Y_1+Y_2$$ will have non-zero skewness. The Levy-Cramér argument does the same sort of thing, only a lot more general.
[If only finitely many $$\mathrm{var}[X_i]$$ are non-zero, the Levy-Cramér argument is much more direct, since the ones with non-zero variance immediately have to be Normal, but that's a special case]