In this post, the user asks whether the following random variable converges to $0$ almost surely:
$X_n = \begin{cases} 0, & \text{with probability 1 - $\frac{1}{2n}$,} \\ n, & \text{with probability $\frac{1}{2n}$} \end{cases}$
In the comments and answers to the question, it is said that it depends on whether $X_n$ is independent or not. But it seems to me that the $X_n$ are independent because since as we have specified their probability distributions, we always know the probability of getting a $0$ or an $n$ for any particular $n$-th event, and it doesn't matter what has happened for previous (or future) events.
One of the answers to the post outlines two cases where we can have, or not have, almost sure convergence:
- Case 1) Doesn't converge: The $X_n$ are independent then the fact that $X_n = n$ infinitely often with probability one follows from the second Borel-Cantelli lemma
- Case 2) Does converge: Let $U \sim$ uniform$(0, 1)$ and set $X_n = n$ if $U < 1 / 2n$. Then our sequence satisfies the conditions of the problem and $X_n \to 0$ almost surely.
Can someone show me how to prove these statements using the definition of independent events $P(A \cup B) = P(A)P(B)$? I'm hoping that if I can see it written out clearly that I can pinpoint where my intuition is wrong.