Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

The observable outputs of many complex dynamical systems consist in time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, algorithms able to generate correlated noises reproducing the properties of real time series produce \textsl{Gaussian} outputs, while real, experimentally observed time series are often non-Gaussian, and may follow distributions with a diversity of behaviors concerning the support, the symmetry or the tail properties. Here, we study how the correlation of two Gaussian variables changes when they are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties, from decays faster than exponential to heavy tail cases including power-laws, and we find how these properties affect the correlation of the final variables. We extend these results to Gaussian time series which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing Gaussian power-law correlated time series in order to create synthetic time series with arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of a real time series: the absolute returns of stock prices.