I'm asked to construct a series of independent random variables $X_k, k \in \mathbb{N}$ all with different distributions, such that
$$N^{-\frac{1}{2}}\sum_{k=1}^NX_k \rightarrow_d X \sim \mathcal{N}(0,1)$$
as $N \rightarrow \infty$. I thought about having them all normal-distributed with expectation 0 and different variance. That the sum of every $2^k, k\in \mathbb{N}$ random variables is again standard normal distributed, but I'm unsure on how to chose the variance. Since they all distributions have to be different.
The variance of the sum of independent random variables is the sum of their variances. So choosing the variances of the $\ X_k\ $ to converge to 1 sufficiently fast as $\ N\rightarrow\infty\ $should work. If you set $\ \mathrm{Var}\left(X_1\right)=\frac{3}{2}\ $ and $\ \mathrm{Var}\left(X_k\right)=1 + \left(2^{-k}-2^{-(k-1)}\right)\ $ for $\ k\ge 2\ $, for instance, then $\ \mathrm{Var}\left(N^{-\frac{1}{2}}\sum_\limits{k=1}^NX_1\right)=1 + \frac{1}{N2^N}\ $.