Kiyono Ken, Konno Hidetoshi
Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 May;87(5):052104. doi: 10.1103/PhysRevE.87.052104. Epub 2013 May 6.
As a possible generalization of Beck-Cohen superstatistical processes, we study non-Gaussian processes with temporal heterogeneity of local variance. To characterize the variance heterogeneity, we define log-amplitude cumulants and log-amplitude autocovariance and derive closed-form expressions of the log-amplitude cumulants for χ(2), inverse χ(2), and log-normal superstatistical distributions. Furthermore, we show that χ(2) and inverse χ(2) superstatistics with degree 2 are closely related to an extreme value distribution, called the Gumbel distribution. In these cases, the corresponding superstatistical distributions result in the q-Gaussian distribution with q=5/3 and the bilateral exponential distribution, respectively. Thus, our finding provides a hypothesis that the asymptotic appearance of these two special distributions may be explained by a link with the asymptotic limit distributions involving extreme values. In addition, as an application of our approach, we demonstrated that non-Gaussian fluctuations observed in a stock index futures market can be well approximated by the χ(2) superstatistical distribution with degree 2.
作为贝克 - 科恩超统计过程的一种可能推广，我们研究具有局部方差时间异质性的非高斯过程。为了刻画方差异质性，我们定义对数幅度累积量和对数幅度自协方差，并推导了卡方分布（χ(2)）、逆卡方分布（inverse χ(2)）和对数正态超统计分布的对数幅度累积量的闭式表达式。此外，我们表明二阶的卡方分布（χ(2)）和逆卡方分布（inverse χ(2)）超统计与一种极值分布密切相关，称为耿贝尔分布。在这些情况下，相应的超统计分布分别导致q = 5/3的q - 高斯分布和双边指数分布。因此，我们的发现提供了一个假设，即这两种特殊分布的渐近出现可能通过与涉及极值的渐近极限分布的联系来解释。此外，作为我们方法的一个应用，我们证明了在股指期货市场中观察到的非高斯波动可以很好地由二阶的卡方分布（χ(2)）超统计分布近似。
