My friend and I have been stumped on this problem for a little while and I thought asking for tips couldn't hurt (we did ask the teacher, but we got other problems after)
Here is the question :
Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables defined on the same probability space $(\Omega, F, \mathbb{P})$ with the same law of finite expected value (E(|X_1|)<\infty ). Let
$Y_n = n^{-1} \max_{1 \leq i \leq n} |X_i|$.
Show that
$\lim_{n\rightarrow \infty} E(Y_n) = 0$
and
$Y_n \rightarrow 0$ almost surely.
We have ideas of many parts of the proof, for example for the first one it would suffice to show that the expected value of the max of all $|X_i|$ is finite... and since the max is one of the $|X_i|$ for each $\omega \in \Omega$ it seems reasonable but we're not sure how to show it.
We also tried splitting the integral for the expected value into a partition of $\Omega$ considering the sets on which $X_i$ is the max, but didn't get too far with that.
For the second part, I think we could show it if we knew that $X_i(\omega)$ diverges for only a measure 0 set, but it's not that obvious (I think).
Any pointers to the right direction appreciated!