Extreme Value Laws for Superstatistics

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Frechet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely chi(2)-superstatistics, inverse chi(2)-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that chi(2)-superstatistics generally leads an extreme value statistics described by a Frechet distribution, whereas inverse chi(2)-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.