Spectral density of the noncentral correlated Wishart ensembles

Abstract

Wishart ensembles of random matrix theory have been useful in modeling positive definite matrices encountered in classical and quantum chaotic systems. We consider nonzero means for the entries of the constituting matrix A which defines the correlated Wishart matrix as W = AA † , and refer to the ensemble of such Wishart matrices as the non-central correlated Wishart ensemble (nc-CWE). We derive the Pastur self-consistent equation which describes the spectral density of nc-CWE at large matrix dimension.

Eigenfunction statistics of Wishart Brownian ensembles

“…multivariate statistical analysis [7], quantum chromodynamics [8], mesoscopic systems [2,9,10], spin glasses [11], financial systems [12,13], information theoretic studies [14] and communications [2,[15][16][17], time-series analysis [18][19][20], biological networks [21], geophysics [22] etc. Based on the nature of constraints on the matrix A (originating from the exact symmetries and conservation laws in the complex system it represents), Wishart ensembles (WE) can be of various types e.g stationary WE [23], correlated WE [18,24,25], beta-WE [26], non-white WE [27], Brownian WE [28]; the present study concerns with last of these type i.e Brownian WE only.…”

Correlations and flow of information between the New York Times and stock markets

“…This kind of matrices have complex eigenvalues. We study the eigenvalues distribution of this matrices in the complex plane by the techniques developed for large dimensions (N → ∞, T → ∞) in [63], but the results (not showed) do not assert the presence of cross-correlations between NYT and the world indices, which might be due to the small dimension (N = 40) of the empirical correlation matrix. To go further we will proceed to study a causality measure instead of linear correlations.…”

The smallest eigenvalue distribution in the real Wishart–Laguerre ensemble with even topology

“…This is particularly true when correlations among the matrix elements are introduced. The introduction of correlations can be done in three ways, either by a correlation matrix resulting in the correlated Wishart ensemble as it is the case in the analysis of real time series [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], by extending to non-Gaussian probability distributions of the matrix elements [19][20][21] or by adding constraints [22][23][24].…”