= Divergence of normalized iid partial sums without a first absolute moment
{title2=$\mathbb E|X_1|=\infty\implies\limsup_n|X_n|/n=\limsup_n|S_n|/n=\infty\text{ a.s.}$}
For independent identically distributed <random variables>, the <tail-sum formula> applied to $|X_1|/\lambda$ shows that $\sum_n\mathbb P(|X_n|\geq\lambda n)=\infty$ 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 $|X_n|/n\leq2C$ 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.
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