So the CLT says the sum of random variables with rapidly decaying tails will approach a normal distribution. There are similar results showing that the sum of slowly decaying random variables approaches a stable distribution:

https://en.wikipedia.org/wiki/Stable_distribution

This makes the stable distribution the right answer under some circumstances.

For different test statistics (e.g. max drawdown), you've got similar fat tailed distributions, e.g. GEV:

https://en.wikipedia.org/wiki/Generalized_extreme_value_dist...

As an example of how you might use slowly decaying distributions, consider this example of Cauchy PCA:

http://arxiv.org/pdf/1412.6506v1.pdf

I'm working on an blog post explaining the use of fat tailed distributions for linear regression in a Bayesian context.