Let $V_1,V_2,\dots$ be i.i.d. $\mathcal{N}(\mu,\sigma)$. How does one find the distribution of $$S_n = \sum_{i=1}^n \log\left(\frac{2}{1+e^{-2V_i}}\right).$$
I can derive the distribution for $S_1$ and the expression is a little cumbersome. I understand that the next step is to convolve this distribution with itself. I am stuck there, I am not able to reduce the expression at all!