Consider the following two statements :
i) Suppose that $X_1, X_2, \dots$ are independent and identically distributed and $E[X_1^-] < \infty, E[X_1^+] = \infty$. Then $n^{-1} \sum_{k=1}^{n}X_k \to \infty$ with probability $1$.
ii) Suppose that $X_1, X_2, \dots$ are independent and identically distributed and $E[|X_1|] = \infty$. It can be proved that $\sup_{n}n^{-1}|\sum_{k=1}^{n}X_k| = \infty$ w.p.$1$.
In a problem it is asked to compare these two statements. What is the meaning of it ?
These two statements give the behavior of $M_n:=S_n/n$ when $X_1$ is not integrable. This means that at least one of the random variables $X_1^+$ or $X_1^-$ is not integrable.
When both are non integrable, the best we can say is that the supremum of the $|M_n|$'s is infinite almost everywhere.
But if we know that $X_1^-$ has a finite expectation, we can say more, namely, $M_n\to+\infty$ as $n\to\infty$ because for each integer $M$ and almost every $\omega$, $$\liminf_{n\to\infty}\frac 1n\sum_{j=1}^nX_j(\omega)\geqslant\mathbb E[X_1^+\chi_{\{X_1^+\leqslant M\}}]-\mathbb E[X_1^-].$$
Here are conditions to determine the conclusion we can expect.