Divergence of normalized iid partial sums without a first absolute moment ($\mathbb{E}|X_1|=\infty ⟹{\mathrm{limsup}}_n|X_n|/n={\mathrm{limsup}}_n|S_n|/n=\infty \text{ a.s.}$)

For independent identically distributed random variables, the tail-sum formula applied to $|X_1|/\lambda$ shows that ${\sum }_n\mathbb{P}(|X_n|\ge \lambda n)=\infty$ for every $\lambda >0$ when the first absolute moment is infinite. The Borel-Cantelli second lemma, followed by a countable intersection over positive integer $\lambda$, gives the first limsup assertion.

If $|S_n|/n$ had finite limsup along such a sample path, it would eventually be bounded by some finite constant $C$. Then $X_n=S_n-S_{n-1}$ would imply $|X_n|/n\le 2C$ eventually, a contradiction. This is a necessary-integrability counterpart to the strong law of large numbers; without a finite first absolute moment, the normalized sums cannot converge to any finite value.