Consider a sequence $(X_n)_{n\ge1}$ of i.i.d random variables with distribution $\mathcal{B}_{1/2}$, and set
$$S_n := X_1 + \cdots + X_n$$
Fix $p \in (0,1),\,q=1-p,$ and define
$$M_n := (2p)^{S_n}(2q)^{n-S_n}$$
$(M_n)_{n\ge1}$ is a martingale w.r.t the filtration generated by $(X_n)_{n\ge1}$. Using the appropriate martingale convergence theorem, show that $(M_n)_{n\ge1}$ admits a.s. limit $M$.
I've already shown that admits a.s limit $M$, but I can't determine it.
Thanks in advance for any help!
The limit is $1$ if $p=q$. This is trivial.
If $p \ne q$, then using $S_n/n \to 1/2$ almost surely (Law of Large number), we have $M_n \to 0$ almost surely.
To see, consider any fixed sequence $s_n$ such that $s_n/n \to 1/2$. It's not difficult to see that $$ (2p)^{s_n} (2(1-p))^{n-s_n} \to 0 $$ unless $p=1/2$.