Tieplova D. Application of large random matrices to multivariate time series analysis

This dissertation is devoted to the study of the behavior of singular empirical values matrices of autocovariances between the past and the future of a high-dimensional Gaussian multivariate time-series and eigenvalues of empirical matrices of covariances with tensor product samples. The asymptotic behavior of the resolvent of the associated Gram matrix of the autocovariance matrix is established. The behavior of the Stieltjes transformation of its deterministic equivalent near the axis of real numbers is studied and the properties for the corresponding boundaries are found. It is established that almost surely, all eigenvalues of the corresponding matrix are located in the neighborhood of the support of the deterministic equivalent measure. It is proved that under the condition of existence of the second moment of vector elements, the normalized counting measure of eigenvalues of covariance matrices with tensor product samples tends weakly almost surely to some non-random measure. An equation that satisfies Stiltier's transformation of the limit measure is also found.