Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

Question

Let $Y_n$ be a sequence of non-negative i.i.d random variables with $EY_n = 1$ and $P(Y_n = 1) < 1$. Consider the martingale process formed by $X_n = \prod_{k=1}^n Y_k$. Use the martingale convergence theorem to show that $X_n \to 0$ almost surely.

I see that the Martingale convergence theorem says that $X_n \to X$ almost surely with $E \lvert X \rvert < \infty$.

I don't see how to reach the conclusion that $X = 0$ or $X_n \to 0$.

I see we can prove that $E \lvert X_n \rvert < \infty$ and that $X_n$ is uniformly integrable and $X_n \to X$ in $L^1$. And that $X_n = E(X \mid \mathcal{F}_n)$.

Answer 1

Since $X_n$ is a positive martingale, it is also a supermartingale bounded below by $0$, therefore $X_n\to X_\infty$ a.s. by supermartingale convergence. Now consider that $P(|Y_n-1|>\varepsilon \textrm{ i.o.})=1$ by Borel-Cantelli II. This implies that $X_\infty=0$ is the only admissible limit rv so that $X_n \to 0$ a.s.

Answer 2

By Jensen's inequality, $b:=E\left(\sqrt{Y_1}\right)<\sqrt{E(Y_1)}=\sqrt{1}=1$. Therefore $E\left(\sqrt{X_n}\right)=b^n\to 0$ as $n\to\infty$. By Fatou's lemma, $E\left(\sqrt{X}\right)=0$.