Levy's 0-1 law and Martinagle

Question

This is from Durrett's probability: Theory and Examples Exercise 4.6.6.

Let $X_n \in [0,1]$ be adpated to $\mathcal{F}_n$. Let $\alpha,\beta>0$ with $\alpha+\beta=1$ and suppose $$\mathbb P(X_{n+1}=\alpha +\beta X_n \mid \mathcal{F}_n)=X_n,\quad \mathbb P(X_{n+1}=\beta X_n \mid \mathcal{F}_n)=1-X_n$$ Show $\mathbb P(\lim_n X_n= 0 \;\text{or}\; 1) =1.$

I know there is question Martingale convergence proof here but can't we prove this using levy's 0-1 law?

What can we say about levy's 0-1 law regarding Martingale Sequence?

Answer 1

$X_n$ is a martingale, and so converges (boundedly) to a random variable $X_\infty$. You'll be able to write a recursion for the second moments $\Bbb E[X_n^2]$, leading, after a passage to the limit as $n\to\infty$, to the conclusion that $\Bbb E[X_\infty^2] = \Bbb E[X_\infty]$.