Almost Sure Convergence for dependent random variables

Question

Let's consider a sequence $X_1,X_2...$ such that $P(X_n=0)=\frac{1}{n}$ and $P(X_n=1)=1-\frac{1}{n}$. Prove that $\{X_n\}$ doesn't converge, almost surely.

So I have solved problem for Independent $\{X_n\}$, but I am struggling to show it for dependent. Can someone help me to prove it or show that isn't right (maybe it's an error in the book) ?

Solution

Answer 1

This is not true in general for dependent random variables. Indeed, let $U$ be a random variable uniformly distributed on $[0,1]$. Define $$X_n = \begin{cases}0&\text{ if $U\leq 1/n$}\\1&\text{ if $U>1/n$}.\end{cases}$$

Then you have that $$\mathbb P(X_n\to 1) = \mathbb P(U>0)=1$$ that is almost sure convergence.