By the Berry--Esseen inequality, the limit distribution of $Z_n^*$ is the standard normal distribution of $a_k=1/k^{1/2}$.
If $a_k=1/2^k$, then the limit distribution is uniform on the interval $[-\sqrt3,\sqrt3]$. This follows (i) because $(X_k+1)/2$ may be viewed as the $k$th binary digit of of a random number uniformly distributed on the interval $[0,1]$ and $Var\,Z_n\to1/3$ as $n\to\infty$.