Fat tails are weird

If you have taken a statistics class it may have included stuff like basic measure theory. Lebesgue measures and integrals and their relations to other means of integration. If your course was math heavy (like mine was) it may have included Carathéodory's extension theorem and even basics of operator theory on Hilbert spaces, Fourier transforms etc. Most of this mathematical tooling would be devoted to a proof of one of the most important theorems on which most of statistics is based - central limit theorem (CLT)

Central limit theorem states that for a broad class of what we in math call random variables (which represent realizations of some experiment which includes randomness), as long as they satisfy certain seemingly basic conditions, their average converges to a random variable of a particular type, one we call normal, or Gaussian. 

The two conditions that these variables need to satisfy are that they are:

  1. Independent
  2. Have finite variance

In human language this means that individual random measurements (experiments) "don't know" anything about each other, and that each one of these measurements "most of the time" sits within a bounded range of values, as in it can actually be pretty much always … Read more...