At a time $0$ a share has a value of $x$ and at a time ${kt \over n}$ $(t >0, k,n \in \mathbb N, k \leq n)$ it has a value of $$X_k :=p^{{tk \over n}} S_1\cdots S_kx,$$ where $p>0$ and the $S_i$ are i.i.d. random variables where $$ P[S_i=1+{\sigma \over \sqrt{n}}]=P[S_i=(1+{\sigma \over \sqrt{n}})^{-1}]={1 \over 2},\, \sigma>0. $$
How can I calculate the characteristic function of $\log S_1$ and $\log X_n$?
Some help here would be much appreciated. Also, how does one show that $\log X_n$ in distribution converges to a normally distributed random variable $Y$ and what is the parameter? Some approaches would be nice.
Ok so I calculated
$$\phi_{\log S_1}(z)={1 \over 2} ((1+{\sigma\over \sqrt{n}})^{iz}+(1+{\sigma\over \sqrt{n}})^{-iz})$$ and $$\phi_{\log X_n}(z)= p^{izt}x^{iz}{1 \over 2^n} ((1+{\sigma\over \sqrt{n}})^{iz}+(1+{\sigma\over \sqrt{n}})^{-iz})^n$$