I am struggling with an exercise… the first part of the exercise was ok, but the second part is a lot less trivial.
$X_n$ are non negative iid random variables.
First part: check that if $E(X_i)$ finite then $\lim X_n/n=0$ almost surely. I showed this using events $A_{k,\epsilon}=\{X_n>n\epsilon\}$, tail formula for $E(X_i)$ and 1st Borel Cantelli lemma.
Second part: on the other hand show that if $E(X_i)=\infty$ then $\operatorname*{p-lim} X_n/n =0 $ (that is $X_n/n \to 0$ in probability) but $P(\limsup X_n/n=\infty)=1$.
I get stuck on this one as it seems to contradict what I learnt in class. I try to use the 2nd Borel Cantelli lemma but can’t get to the conclusion.
For the first half: let $X$ be a fixed random variable with the same distribution as the $X_n$ (you can take $X=X_1$ if you like). Then $P(X_n/n \ge \epsilon) = P(X/n \ge \epsilon) = P(X \ge \epsilon n)$. Show that this converges to $0$ as $n \to \infty$. Hint: what can you say about $\bigcap_{n =1}^{\infty} \{X \ge \epsilon n\}$?
For the second half, start by showing the following lemma: if $X$ is any nonnegative random variable, then $$E[X]-1 \le \sum_{n=1}^\infty P(X \ge n) \le E[X]$$ Hint: show $(X-1) \le \sum_{n=1}^\infty 1_{X \ge n} \le X$ for each $\omega$, by considering what happens for each side when $k-1 \le X(\omega) < k$ for a given integer $k$.
Use this with the second Borel-Cantelli lemma to show that $P(X_n \ge n \text{ i.o.}) = 1$, which says that $\limsup X_n/n \ge 1$ almost surely. Now repeat, replacing $X_n$ with $X_n/m$ for arbitrary $m$, to get $\limsup X_n/n \ge m$ a.s. Finally take a countable intersection to get $\limsup X_n/n = \infty$ a.s.