Correlated Wishart ensembles and chaotic time series

We study the correlated Wishart ensembles in the context of time series analysis. We are interested in the statistics of eigenlevels, viz. variables associated with independent eigenmodes in the system. The motivation of this work is to study the effect of time series correlations on the Wishart ensembles. In this connection, we derive the level density and the two-point function for the correlated Wishart ensembles by using the binary correlation method. Using our analytic results we analyze spectra of autocovariance matrices derived from single variable stationary time series. We consider the stochastic time series of Gaussian variables with exponentially decaying correlations and time series of chaotic maps, viz. the Arnold map, the Standard map and the stadium billiard map. In both cases, correlated time series are encountered and analyzed under the framework of random matrix theory. It is shown that the eigenlevel statistics for the chaotic maps follow closely those of correlated Wishart ensembles. It is indicated that the presence of collective modes in the spectra of autocovariance matrices is related to the integrability of the system.