Log-amplitude cumulants and parameter estimation for Beck-Cohen superstatistics

To characterize non-Gaussian stochastic processes, we introduce log-amplitude cumulants which is capable of quantifying systematic deviations from a Gaussian distribution. The advantages of the log-amplitude statistics are that even in the case of heavy tailed distributions with infinite variance, such as Lévy stable distributions and q-Gaussian distributions with q > 5/3, all the log-amplitude cumulants take finite values, and that these statistics can be easily estimated from observed time series. As an application of our approach, we derive closed-form expressions of log-amplitude cumulants for non-Gaussian distributions appearing in Beck-Cohen superstatistics based on gamma, inverse gamma, log-normal and F-distributions. In addition, we propose parameter estimation method for the superstatistical distributions.