This is a purely explorative question.
I asked a question here about a "central limit theorem" for random variables with infinite variance.
I did not expect it, but it turns out that even some random variables with infinite variance nevertheless converge in distribution to a "stable distribution" (although not the normal distribution).
This makes me wonder about something which seems even more implausible: Is there a kind of "law of large numbers" for random variables with an infinite variance?
It does not necessarily have to state that the sample mean converges in probability to the expectation, but perhaps there is something that comes close.