Martingale convergence theorem for the Bernoulli distribution

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 $$p=1/2$$.
• Here we have $S_n$ and not $S_n/n$ though. – Math1000 Dec 11 '19 at 20:40
• @Math1000 We don't need $S_n$ to converge. Just replace $S_n$ by $(1+o(1))n/2$. This is enough to make the sequence converge to $0$. – ablmf Dec 11 '19 at 20:42