Heavy tails are ubiquitous in statistical modelling, however there are a few mechanisms giving birth to them. Here is a non-comprehensive list.
## Simple transformations
Suppose that is a random variable with a continuous density with . Then, will be heavy tailed.
Kesten's theorem is an absolute gem in mathematics and probability. Roughly speaking, it says that if a series of random numbers is defined by the recursion where are random variables independent of , then the limit of has a heavy tail with index given by the equation .
The Fisher-Tippett-Gnedenko theorem says that if are iid random variables and if there are numbers such that converges in distribution, then either the limit is a Gumbel distribution, or it is heavy-tailed (Weibull or Fréchet).
Extremes of random walks are usually heavy-tailed, which is not so surprising given Kesten's theorem.
Log-normal distributions can arise in financial models; under the hypothesis that returns follow a drifted Brownian motion, Ito's formula says that the price at time follows a log-normal distribution. More generally, if a time-varying quantity has a growth rate which does not depend on , then it is heavy-tailed: for example if with the being iid, then which by the CLT is approximately with , which is log-normal.
Zipf's law is one of the most famous heavy-tailed distributions "from the real world".
Real-world graphs often have heavy-tailed degree sequences. A very beautiful explanation of this phenomenon lies in the famous scale-free property of preferential attachment mechanisms. In PA models, new elements (people, requests, molecules) arrive at each time step; when a new element arrives, it connects to (say) older elements, but it favors elements which alrealy have many connections. Remco van der Hofstadts's book has a whole chapter explaining why the degree of elements in such a model are asymptotically heavy-tailed.