  Extreme events and the heavy tail distributions driven by them are ubiquitous
in various scientific, engineering and financial research. They are typically
associated with stochastic instability caused by hidden unresolved processes.
Previous studies have shown that such instability can be modeled by a
stochastic damping in conditional Gaussian models. However, these results are
mostly obtained through numerical experiments, while a rigorous understanding
of the underlying mechanism is sorely lacking. This paper contributes to this
issue by establishing a theoretical framework, in which the tail density of
conditional Gaussian models can be rigorously determined. In rough words, we
show that if the stochastic damping takes negative values, the tail is
polynomial; if the stochastic damping is nonnegative but takes value zero, the
tail is between exponential and Gaussian. The proof is established by
constructing a novel, product-type Lyapunov function, where a Feynman-Kac
formula is applied. The same framework also leads to a non-asymptotic large
deviation bound for long-time averaging processes.
