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Exact multivariate amplitude distributions for non-stationary Gaussian or algebraic fluctuations of covariances or correlations. (English) Zbl 1519.62020

Summary: Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Extending previous work, we view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. The results are either of closed forms or single integrals. We thus provide, first, explicit multivariate distributions for such non-stationary systems and, second, a tool that quantitatively captures the degree of non-stationarity in the correlations.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
60G10 Stationary stochastic processes
60G15 Gaussian processes
62H10 Multivariate distribution of statistics
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