  A key feature of the classical Fluctuation Dissipation theorem is its ability
to approximate the average response of a dynamical system to a sufficiently
small external perturbation from an appropriate time correlation function of
the unperturbed dynamics of this system. In the present work, we examine the
situation where the state of a nonlinear dynamical system is perturbed by a
finitely large, instantaneous external perturbation (jump) -- for example, the
Earth climate perturbed by an extinction level event. Such jump can be either
deterministic or stochastic, and in the case of a stochastic jump its
randomness can be spatial, or temporal, or both. We show that, even for large
instantaneous jumps, the average response of the system can be expressed in the
form of a suitable time correlation function of the corresponding unperturbed
dynamics. For stochastic jumps, we consider two situations: one where a single
spatially random jump of a system state occurs at a predetermined time, and
another where jumps occur randomly in time with small space-time dependent
statistical intensity. For all studied configurations, we compute the
corresponding average response formulas in the form of suitable time
correlation functions of the unperturbed dynamics. Some efficiently computable
approximations are derived for practical modeling scenarios.
