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 of 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, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the 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. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables 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 a 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 a Gaussian power-law correlated time series in order to create a synthetic time series with an 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 real time series: the absolute returns of stock prices.