Calculate convergence of random variables

Question

We are given $X_1,X_2,...$which are all independent random variables and have $Exp(\ln n)$ distribution. Our task is to show that this random variables converge to 0 with probability but not almost surely. I am hitting a wall with this one. What should be my approach here?

Answer 1

Hint: Recall the definition of convergence in probability: $X_n \xrightarrow{p} 0$ means that for each $\epsilon > 0$, $\mathbb{P}(X_n > \epsilon) \to 0$ as $n \to \infty$. Can you compute $\mathbb{P}(X_n > \epsilon)$ explicitly?

To show that there isn't almost sure convergence, show that for some $\epsilon > $, $\mathbb{P}(X_n > \epsilon \text{ infinitely often}) = 1$. How do you show that something happens infinitely often?