Athreya showed that the bootstrap distribution of a sum of infinite variance random variables did not (with probability 1) tend weakly to a fixed distribution but instead tended in distribution to a random distribution. In this paper, we give a different proof of Athreya's result motivated by a heuristic large sample representation of the bootstrap distribution.
"On the Bootstrap of the Sample Mean in the Infinite Variance Case." Ann. Statist. 17 (3) 1168 - 1175, September, 1989. https://doi.org/10.1214/aos/1176347262