  We study the stability of deterministic systems given sequences of large,
jump-like perturbations. Our main result is to dervie a lower bound for the
probability of the system to remain in the basin, given that perturbations are
rare enough. This bound is efficient to evaluate numerically.
  To quantify rare enough, we define the notion of the independence time of
such a system. This is the time after which a perturbed state has probably
returned close to the attractor, meaning that subsequent perturbations can be
considered separately. The effect of jump-like perturbations that occur at
least the independence time apart is thus well described by a fixed probability
to exit the basin at each jump, allowing us to obtain the bound.
  To determine the independence time, we introduce the concept of finite-time
basin stability, which corresponds to the probability that a perturbed
trajectory returns to an attractor within a given time. The independence time
can then be determined as the time scale at which the finite-time basin
stability reaches its asymptotic value. Besides that, finite-time basin
stability is a novel probabilistic stability measure on its own, with potential
broad applications in complex systems.
