Extreme Value Laws for Superstatistics

Abstract

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, Fréchet 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 Fréchet distribution, whereas inverse ${\chi }^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.

To appear in Entropy

References in corpus (12)

• Defect turbulence and generalized statistical mechanics
• Asymptotics of superstatistics
• Statistics of 3-dimensional Lagrangian turbulence
• Return interval distribution of extreme events and long term memory
• Superpositions of Probability Distributions
• Measurement Invariance, Entropy, and Probability
• Hierarchical maximum entropy principle for generalized superstatistical systems and Bose-Einstein condensation of light
• Heterogeneous anomalous diffusion in view of superstatistics
• Statistics of Lagrangian quantum turbulence
• Superstatistical analysis of sealevel fluctuations
• Statistical mixing and aggregation in Feller diffusion
• Upper Entropy Axioms and Lower Entropy Axioms