  We investigate Lee-Yang zeros of generating functions of dynamical
observables and establish a general relation between phase transitions in
ensembles of trajectories of stochastic many-body systems and the time
evolution of high-order cumulants of such observables. This connects dynamical
free-energies for full counting statistics in the long-time limit, which can be
obtained via large-deviation methods and whose singularities indicate dynamical
phase transitions, to observables that are directly accessible in simulation
and experiment. As an illustration we consider facilitated spin models of
glasses and show that from the short-time behavior of high-order cumulants it
is possible to infer the existence and location of dynamical or "space-time"
transitions in these systems.
