Estimator of a non-Gaussian parameter in multiplicative log-normal models

We study non-Gaussian probability density functions (PDF's) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing's model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the q th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF's observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.