  We study heavy tailed random variables obtained by sampling an observable
with a power singularity along an orbit of an expanding map on the circle. In
the regime where the observables have infinite variance, we show that the set
of limit points of the ergodic sum process, rescaled by $N^{1/s} (\ln
N)^\alpha$, where $s$ is the index of limiting stable law, equals to the set of
piecewise constant increasing functions with finitely many jumps. The proof
relies on the multiple Borel-Cantelli Lemma proven in Dolgopyat, Fayad and Liu
[J. Mod. Dyn. 18 (2022) 209--289].
