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Log-amplitude statistics for Beck-Cohen superstatistics

¹Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan
²Department of Risk Engineering, Faculty of Information and Systems, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan

Phys. Rev. E 87, 052104 – Published 6 May, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.052104

Abstract

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.

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Log-amplitude statistics for Beck-Cohen superstatistics

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