Q:

Sum of Random Variables i.i.d. with $\mathbb{E}[|X_n|]=+\infty$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $\mathbb{E}[|X_n|]=+\infty,\forall n$. Prove that
$\sum_n \mathbb{P}[|X_n|>kn]=\infty$ with $k\in \mathbb{N}$

A:

Hint:
$ +\infty = \mathbb{E}[|X_n|] = \int_0^{+\infty} \mathbb{P}[|X_n > y|] \ dy =  \sum_{n=0}^{+\infty} \int_n^{n+1} \mathbb{P}[|X_n| >  y] \ dy \leq \sum_{n=0}^{+\infty} \int_n^{n+1} \mathbb{P}[|X_n| > n] dy = \sum_{n=0}^{+\infty}  \mathbb{P}[|X_n| > n] \int_n^{n+1} 1 dy = \sum_{n=0}^{+\infty}  \mathbb{P}[|X_n| > n]  $

