Limit supremum of i.i.d. sequence

Question

Let $X_n$ be i.i.d. sequence of random variables with $E|X_n| = \infty$. I showed, \begin{equation} \sum_n P(|X_n| > kn) = \infty \text{ for all } k\geq 1 \end{equation} and \begin{equation} \limsup_n \frac{|X_n|}{n} = \infty\quad a.s. \end{equation} Now, from here I need to claim that \begin{equation} \limsup_n \frac{|\sum_k^n X_k|}{n} = \infty \quad a.s. \end{equation} I am not sure how pass to summation from the result for each term. I suspect it is something standard that I am missing.

Answer 1

Let $S_n = \sum\limits_{k=1}^n X_k$. Then we can write $$ \frac{S_n}{n} = \frac{X_n}{n} + \frac{S_{n-1}}{n-1} \frac{n-1}{n}. $$ Now if we assume $\mathbb{P}( \limsup \frac{|S_n|}{n} < \infty) > 0 $ this means that the sequence $\frac{S_n}n$ is bounded with positive probability. Hence, so is $\frac{X_n}{n}$ from the above equality, which is a contradiction.