  In many real world chaotic systems, the interest is typically in determining
when the system will behave in an extreme manner. Flooding and drought, extreme
heatwaves, large earthquakes, and large drops in the stock market are examples
of the extreme behaviors of interest. For clarity, in this paper we confine
ourselves to the case where the chaotic system to be predicted is stationary so
theory for asymptotic consistency can be easily illuminated. We will start with
a simple case, where the attractor of the chaotic system is of known dimension
so the answer is clear from prior work. Some extension will be made to
stationary chaotic system with higher dimension where a number of empirical
results will be described and a theoretical framework proposed to help explain
them.
