  We study the connection between the cumulants of a time-integrated observable
of a quantum system and the PT-symmetry properties of the non-Hermitian
deformation of the Hamiltonian from which the generating function of these
cumulants is obtained. This non-Hermitian Hamiltonian can display regimes of
broken and of unbroken PT-symmetry, depending on the parameters of the problem
and on the counting field that sets the strength of the non-Hermitian
perturbation. This in turn determines the analytic structure of the long-time
cumulant generating function (CGF) for the time-integrated observable. We
consider in particular the case of the time-integrated (longitudinal)
magnetisation in the one-dimensional Ising model in a transverse field. We show
that its long-time CGF is singular on a curve in the magnetic field/counting
field plane that delimits a regime where PT-symmetry is spontaneously broken
(which includes the static ferromagnetic phase), from one where it is preserved
(which includes the static paramagnetic phase). In the paramagnetic phase,
conservation of PT -symmetry implies that all cumulants are sub-linear in time,
a behaviour usually associated to the absence of decorrelation.
