  The cumulant representation is common in classical statistical physics for
variables on the real line and the issue of closures of cumulant expansions is
well elaborated. The case of phase variables significantly differs from the
case of linear ones; the relevant order parameters are the Kuramoto-Daido ones
but not the conventional moments. One can formally introduce `circular'
cumulants for Kuramoto-Daido order parameters, similar to the conventional
cumulants for moments. The circular cumulant expansions allow to advance beyond
the Ott-Antonsen theory and consider populations of real oscillators. First, we
show that truncation of circular cumulant expansions, except for the
Ott-Antonsen case, is forbidden. Second, we compare this situation to the case
of the Gaussian distribution of a linear variable, where the second cumulant is
nonzero and all the higher cumulants are zero, and elucidate why keeping up to
the second cumulant is admissible for a linear variable, but forbidden for
circular cumulants. Third, we discuss the implication of this truncation issue
to populations of quadratic integrate-and-fire neurons [E. Montbri\'o, D.
Paz\'o, A. Roxin, Phys. Rev. X, vol. 5, 021028 (2015)], where within the
framework of macroscopic description, the firing rate diverges for any finite
truncation of the cumulant series, and discuss how one should handle these
situations. Fourth, we consider the cumulant-based low-dimensional reductions
for macroscopic population dynamics in the context of this truncation issue.
These reductions are applicable, where the cumulant series exponentially decay
with the cumulant order, i.e., they form a geometric progression hierarchy.
Fifth, we demonstrate the formation of this hierarchy for generic distributions
on the circle and experimental data for coupled biological and electrochemical
oscillators.
