How to compute the almost sure limit in Martingale Convergence Theorem

Question

I have showed that $M_n = e^{S_n-n/2}$ is a martingale w.r.t $\mathcal{F}_n=\sigma\{X_1, X_2,...,X_n\}$, where $X_k$ are all i.i.d standard normals. So the MCT tells us that $M_n$ has almost sure limit because $EM_n^+ = 1.$

But how can I identify the limit? Tried using SSLN, but that does not seem to apply here.

Answer 1

Just use $S_n \sim \mathcal N(0,n)$ here. Then it is easy to show that the exponent converges to $-\infty$ in distribution (for example). So the entire sequence converges to 0 in distribution. Since you have already shown convergence a.s., 0 is also the a.s. limit.