Multivariate Distributions in Non-Stationary Complex Systems I: Random Matrix Model and Formulae for Data Analysis

Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant variables, especially in the presence of non-stationarity. We use a generalized scalar product between correlation matrices to clearly demonstrate this non-stationarity. Further, we present a model that we recently put forward, which captures how the non-stationary fluctuations of correlations make the tails of multivariate distributions heavier. Here, we provide the resulting formulae including Gaussian or Algebraic features. Compared to our previous results, we manage to remove in the Algebraic cases one out of the two, respectively three, fit parameters which considerably facilitates applications. We demonstrate the usefulness of these results by deriving joint distributions for linear combinations of amplitudes and validating them with financial data. Furthermore, we explicitly work out the moments of our model distributions. In a forthcoming paper we apply the model to financial markets.