We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance (correlation) matrix. Results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom, like for instance those in statistical physics, lattice measurements of field theory, genetics, quantitative finance and other applications of multivariate statistics.

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We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.Portfolio theory, Power laws, Statistical physics, Risk measures, Random walks, Options pricing, Random matrix theory,

This is a longer version of our article Burda et al., Phys. Rev. E82, 061114 (2010), containing more detailed explanations and providing pedagogical introductions to the methods we use. We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so-called M -transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.