超统计量的极值律
Extreme Value Laws for Superstatistics
摘要：
研究了超统计量模型下随机过程的极值分布。经典极值理论断言（在温和的渐近独立假设下）只有三种可能的极限分布是可能的，即：Gumbel，Fréchet和Weibull分布。另一方面，超统计包含三个重要的普适类，即$χ^2$ -超统计、逆$χ^2$ -超统计和对数正态超统计，所有这些都最大化不同的有效熵度量。我们研究了三类极值理论与三类超统计量之间的关系。我们表明，任何超统计过程的局部平衡分布不生活在一个有限的支持，威布尔分布不能发生。在上述温和的渐近独立性假设下，我们还表明，$χ^2$ -超统计量通常会导致一个极值统计描述的Fréchet分布，而逆$χ^2$ -超统计量，以及对数正态超统计量，导致一个极值统计与Gumbel分布。
英文摘要：
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 $χ^2$-superstatistics, inverse $χ^2$-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 $χ^2$-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse $χ^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.
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