Exact multivariate amplitude distributions for non-stationary Gaussian or algebraic fluctuations of covariances or correlations

Thomas Guhr,Andreas Schell

doi:10.1088/1751-8121/abe3c8

Abstract

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.

Highlights

A wealth of complex systems show non–stationarity as characteristic feature, i.e. they lack any kind of equlibrium [1, 2, 3, 4]
Most approaches of statistical physics based on the existence of equilibrium, stationarity or quasi–stationarity are not applicable, these systems pose questions similar to the ones in equlibrium
Universal behavior and how can we identify it? — What are proper statistical models, i.e. models based on the assumption of randomness for some parts or aspects of the systems? — Sometimes other approaches, e.g. from many–body physics carry over or provide useful inspiration

Summary

Introduction

A wealth of complex systems show non–stationarity as characteristic feature, i.e. they lack any kind of equlibrium [1, 2, 3, 4]. The time series of wave intensities at different positions change with the direction or the composition of the wave packet which modifies the correlations [9, 10, 11]. Finance features this type of non–stationarity because the business relations between the firms and the traders’ market expectations change, the non–stationarity becomes dramatic in the state of crisis [12, 13, 14, 15, 16, 17, 18]. This is especially so for the interplay between coherent, collective motion and incoherent motion of the individual particles [23, 24, 25]

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Conclusion