Suppose $W_1,W_2,....$ are independent and identically distributed random variables such that $P(W_1=0)=P(W_1=2) =1/2$. For each $n=1,2,3...$ define the random variable $X_n = W_1.W_2...W_n$. Show that
a) $X_n \rightarrow 0$ a.s as $n \rightarrow \infty$.
b) $X_n$ fails to converge to $0$ in $\mathcal{L}_1$ as $ n \rightarrow \infty$?
For part a), firstly I showed that the random variable $X_n$ is a martingale and $\sup_n E(|X_n|)< \infty$.
Then $X_n$ converges to $X$ as $n \rightarrow \infty$ $a.s$. To show that $X=0$, By strong law of large numbers, $$\log(X_n)= \log\left(W_1.W_2....W_n\right)=n\left(\frac{1}{n}\sum_{i=1}^n\log(W_i)\right) \rightarrow -\infty~~ a.s$$ This implies that $X_n \rightarrow 0$ as $n \rightarrow \infty$ a.s.
For part c) I need to show that $X_n$ is not uniform integrable. I am not sure how to start this problem. Can anyone give me some hint for this part?
Is the solution of part b) is correct?
$X_n\to 0$ a.s. because $\mathsf{P}(W_n=0 \text{ for some }n\ge 1)=1$. (In fact, $W_n=0$ i.o. a.s.) Now, the uniform integrability of $\{X_n\}$ would imply the $L^1$ convergence, and the latter fails by part (b).