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

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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 $n\to\infty$. 

For the more general case with $a_k=t^k$ for $t\in(0,1)$, see e.g. [this survey][2]. 

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Using the formula (cf. e.g. [formula 1.411.6 for $\tanh=(\ln\circ\cosh)'$][3])
$$\ln\cosh x=\sum_{j=1}^\infty \frac{2^{2 j-1} \left(4^j-1\right) B_{2 j}}{j (2 j)!}\,x^{2j}$$
for all $x\in(-\pi/2,\pi/2)$, where the $B_{2 j}$'s are the Bernoulli numbers, we get all the [cumulants][4] and thence the [moments][5] of $Z_n$, in terms of [Bell polynomials][6]. 


  [1]: https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands
  [2]: https://arxiv.org/abs/1608.04210
  [3]: http://fisica.ciens.ucv.ve/~svincenz/TISPISGIMR.pdf
  [4]: https://en.wikipedia.org/wiki/Cumulant#Definition
  [5]: https://en.wikipedia.org/wiki/Cumulant#Cumulants_and_moments
  [6]: https://en.wikipedia.org/wiki/Bell_polynomials#Bell_polynomials