= Rare-jump counterexample to a strong law with convergent first-moment averages
{title2=$\mathbb P(X_n=1\pm n)=1/(2n\log n),\quad X_n\to1\text{ in }L^1,\quad n^{-1}\sum_{i=2}^{n+1}X_i\not\to\text{ a finite limit a.s.}$}
Take independent <random variables> equal to $1+n$ and $1-n$ with probability $1/(2n\log n)$ each, and otherwise equal to one. Their first absolute deviations are $1/\log n\to0$, so $X_n\to1$ in <L1 convergence>. The same triangle-inequality argument gives convergence in $L^1$ of their arithmetic averages to one. For $p>1$, however, $\mathbb E|X_n-1|^p=n^{p-1}/\log n\to\infty$.
The exceptional jumps occur infinitely often almost surely by the <Borel-Cantelli second lemma>, since $\sum1/(n\log n)=\infty$. If the arithmetic averages $S_n/n$ converged to any finite value along a sample path, then $X_{n+1}/n=S_n/n-[(n-1)/n][S_{n-1}/(n-1)]$ would tend to zero. At each exceptional jump its magnitude is at least one. Thus the averages have no finite almost sure limit. Independence and convergence of first-moment averages do not alone give a <strong law of large numbers>.
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