By the [Berry--Esseen inequality][1], the limit distribution of $Z_n^*$ is the standard normal distribution if $a_k=1/k^{1/2}$. 

--- 

If $a_k=1/2^k$, then the limit distribution of $Z_n^*$ 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 a random number uniformly distributed on the interval $[0,1]$ and (ii) $Var\,Z_n\to1/3$ as $n\to\infty$.

---

Using the formula 
$$\ln\cosh x=\sum_{j=1}^\infty \frac{2^{2 j-1} \left(4^j-1\right) B_{2 j}}{j (2 j)!}\,x^{2j}$$
for all $x\in(-\pi/2,\pi/2)$, where the $B_{2 j}$'s are the Bernoulli numbers, we get all the [cumulants][2] and thence the [moments][3] of $Z_n$, in terms of [Bell polynomials][4]. 


  [1]: https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands
  [2]: https://en.wikipedia.org/wiki/Cumulant#Definition
  [3]: https://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
  [4]: https://en.wikipedia.org/wiki/Bell_polynomials#Bell_polynomials