Nice piece of research here, and almost more so the generated discussion within the commentary.

If I may weigh in a few of the raised issues.

The term of your returns will most certainly have a large impact on the obtained return distribution. And, unfortunately, this need not approach normality. An oft quoted but somewhat misinterpreted stylized market fact a al Cont (2001) is that of aggregational guassianity, with the rule of thumb being that returns can be considered close enough to normality from 1M onwards (Bingham & Kiesel (2004)). This is definitely not true in certain markets, under both rolling and resampled x-month returns. In your case, your dataset is large enough to choose independent periods, thus alleviating any autocorrelation issues.

In addition, depending on the period chosen, one will find quite severely differing results, even if the chosen periods are somewhat overlapping.

That said, what one can posit is that returns are ergodic. However, ergodicity is very difficult to

In terms of non-linear risk measures, I would suggest coupling this type of analysis with your simple VaR and CVaR measures. The nice point here is that due to the size of your dataset, you can quite easily use a kernel density estimator to find the specified percentile and mean below that with without having to worry too much about estimation/sample size effects here.

I would also suggest considering Omega. If you are not aware, there is a great picture of two extremely different distributions superimposed with the statement: 'these distributions have the same mean and variance'. This would capture the potential differences between negative and positive 'fat tail' events, and would also allow you to quantify with a bit more rigour what 'fat tail' really means and the extent of its effects.

Finally, in terms of the statement that a negative fat tailed event is most often followed by a positive fat tailed event - I am not so sure. One is uniquely aware at a base level of the gain/loss asymmetry within returns which immediately points to there being more negative extreme events than corresponding positives. However, in order to properly analyse this type of statement, one should really make use of survival (reliability) analysis techniques. While typical survival analysis models the time until 'death' of a population for example, one can quite easily define survival as being within certain sigma bounds and 'death' being an extreme value. Thus one can accurately capture the dynamics of the recurrence times between extreme negative events or between extreme positive events and more importantly, the recurrence time between moving from a neg (pos) extreme to a pos (neg) extreme. In essence, one would focus on the probability of moving from one extreme event to the next extreme (fore example, down-to-up), conditional upon past survival (no extremes). The hazard function considers exactly this.

I am always surprised by how under-utilised this type of analysis is in financial research.

Yours in research,
Emlyn