When they say "fat end of the tail," a Gaussian distribution is not necessarily implied. They are simply saying that when you go to the extreme ends of any distribution, almost any statistical model will begin to fall apart. If you look normalized data and look at a Q-Q plot, you will no doubt see problems at the extremes of the distribution, making those predictions more difficult.

As for the Omori distribution, I do not know anything about it, but I have studies similar distributions for predicting future Olympic running records,tallest human alive, etc, and these types of distributions rarely produce practically feasible results. If this model works well, I will be thoroughly impressed.

The only field that I know which deals with extreme events is Ruin Theory, but it is currently a very limited field. It may be possible for someone to adapt the field to study Macroeconomics, but even that may not be very informative.

At humanities current understanding of economics, I would argue that Black Swan Theory is the only practical way to understand huge economic shifts. Perhaps we will understand economic markets well enough to develop more complicated models, but that seems far in the future.

Q-Q plot: http://en.wikipedia.org/wiki/Q-Q_plot Ruin Theory: http://www.worldscibooks.com/etextbook/5943/5943_chap01.pdf