  We characterize the complex, heavy-tailed probability distribution functions
(pdf) describing the response and its local extrema for structural systems
subjected to random forcing that includes extreme events. Our approach is based
on the recent probabilistic decomposition-synthesis technique, where we
decouple rare events regimes from the background fluctuations. The result of
the analysis has the form of a semi-analytical approximation formula for the
pdf of the response (displacement, velocity, and acceleration) and the pdf of
the local extrema. For special limiting cases (lightly damped or heavily damped
systems) our analysis provides fully analytical approximations. We also
demonstrate how the method can be applied to high dimensional structural
systems through a two-degrees-of-freedom structural system undergoing rare
events due to intermittent forcing. The derived formulas can be evaluated with
very small computational cost and are shown to accurately capture the
complicated heavy-tailed and asymmetrical features in the probability
distribution many standard deviations away from the mean, through comparisons
with expensive Monte-Carlo simulations.
