Michael,
Thanks for the comment, This paper has intrigued me since I heard Mikosch present it at the Extreme Value Theory Conference in Colorado in 2009. I can visualize it working in the domain of attraction of a stable distribution, but the proof gets complicated when you get to lighter power tails and how they should be scaled to have the limit work, and I am not sure that I fully understand it.
If you take a lighter power tail, say a StudentTDistribution[3], with summation the power tail behavior when alpha = 3 moves further and further out on the distribution, until it becomes unlikely to ever happen, I have not been able to simulate the evolution to a Fréchet type of extreme value distribution with various scalings of a StudentTDistribution[3] random walk. The good fit of the Log[hi/lo] measure to the MaxStable/distribution is interesting and hard to explain when you look at higher frequency price data. The strong serial dependence of the measure should be a warning that this is not an independent random variable.
The measure tells us something about the intraday trading, and it can be applied to individual stocks with the same results, but how it evolves is a mystery to me.