Abstract:
We present an analytic method to determine spectral properties of the covariance matrices constructed of correlated Wishart random matrices. The method gives, in the limit of large matrices, exact analytic relations between the spectral moments and the eigenvalue densities of the covariance matrices and their estimators. The results can be used in practice to extract information about the genuine correlations from the given experimental realization of random matrices.

Abstract:
Free probability and random matrix theory has shown to be a fruitful combination in many fields of research, such as digital communications, nuclear physics and mathematical finance. The link between free probability and eigenvalue distributions of random matrices will be strengthened further in this paper. It will be shown how the concept of multiplicative free convolution can be used to express known results for eigenvalue distributions of a type of random matrices called Information-Plus-Noise matrices. The result is proved in a free probability framework, and some new results, useful for problems related to free probability, are presented in this context. The connection between free probability and estimators for covariance matrices is also made through the notion of free deconvolution.

Abstract:
We study {the} complex eigenvalues of the Wishart model defined for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as $\mathsf{C}=\mathsf{AB}^{t}/T$, where $\mathsf{B}^{t}$ is the transpose of $\mathsf{B}$ and both matrices $\mathsf{A}$ and $\mathsf{B}$ are of dimension $N\times T$. We consider {\it actual} correlations between the matrices so that on the ensemble average $\mathsf{C}$ does not vanish. We derive a loop equation for the spectral density of $\mathsf{C}$ in {the} large $N$ and $T$ limit where the ratio $N/T$ is finite. The actual correlations changes the complex eigenvalues of $\mathsf{C}$, and hence their domain from the results known for the vanishing $\mathsf{C}$ or for the uncorrelated $\mathsf{A}$ and $\mathsf{B}$. Using the loop equation we derive {a} result for the contour describing the domain of {the} bulk of the eigenvalues of $\mathsf{C}$. If the nonvanishing-correlation matrix is diagonal with the same element $c\ne0$, the contour is no longer a circle centered at origin but a shifted ellipse. In this case, the loop equation is analytically solvable and we explicitly derive {a} result for the spectral density. For more general cases, our analytical result implies that the contour depends on its symmetric and anti-symmetric parts if the nonvanishing-correlation matrix is nonsymmetric. On the other hand, if it is symmetric then the contour depends only on the spectrum of the correlation matrix. We also provide numerics to justify our analytics.

Abstract:
We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered.

Abstract:
The Wishart model for real symmetric correlation matrices is defined as $\mathsf{W}=\mathsf{AA}^{t}$, where matrix $\mathsf{A}$ is usually a rectangular Gaussian random matrix and $\mathsf{A}^{t}$ is the transpose of $\mathsf{A}$. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices $\mathsf{A}$ and $\mathsf{B}$ as $\mathsf{AB}^{t}$. The corresponding Wishart model, thus, is defined as $\mathbf{C}=\mathsf{AB}^{t}\mathsf{BA}^{t}$. We study the spectral density of $\mathbf{C}$ for the case when $\mathsf{A}$ and $\mathsf{B}$ are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of large $\mathbf{C}$. We complement our analytic results with numerics.

Abstract:
By using a symbolic method, known in the literature as the classical umbral calculus, the trace of a non-central Wishart random matrix is represented as the convolution of the trace of its central component and of a formal variable involving traces of its non-centrality matrix. Thanks to this representation, the moments of this random matrix are proved to be a Sheffer polynomial sequence, allowing us to recover several properties. The multivariate symbolic method generalizes the employment of Sheffer representation and a closed form formula for computing joint moments and cumulants (also normalized) is given. By using this closed form formula and a combinatorial device, known in the literature as necklace, an efficient algorithm for their computations is set up. Applications are given to the computation of permanents as well as to the characterization of inherited estimators of cumulants, which turn useful in dealing with minors of non-central Wishart random matrices. An asymptotic approximation of generalized moments involving free probability is proposed.

Abstract:
We consider the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a distinct given covariance matrix. Such a setup appears frequently in multivariate statistics and enjoys various applications. Only in the degenerate case of two equal covariance matrices H reduces to a single rectangular correlated Wishart matrix, whose spectral statistics is known. Our starting point are recent results by Kumar for the distribution of the matrix H valid in the non-degenerate case. It is given by a confluent hypergeometric function of matrix argument. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the joint probability density of the eigenvalues of H reduces to a bi-orthogonal ensemble containing ordinary hypergeometric functions. We compute all spectral k-point density correlation functions of H for arbitrary size N. In the half-degenerate case they are given by a determinant of size k of a kernel of certain bi-orthogonal functions. The latter follows from computing the expectation value of a single characteristic polynomial. In the non-degenerate case using superbosonisation techniques we compute the generating function for the k-point resolvent given by the expectation value of ratios of characteristic polynomials.

Abstract:
In this work we study the spectral density of products of Wishart diluted random matrices of the form $X(1)\cdots X(M)(X(1)\cdots X(M))^T$ using the Edwards-Jones trick to map this problem into a system of interacting particles with random couplings on a multipartite graph. We apply the cavity method to obtain recursive relations in typical instances from which to obtain the spectral density. As this problem is fairly rich, we start by reporting in part I a lengthy analysis for the case of dense matrices. Here we derive that the spectral density is a solution of a polynomial equation of degree $M+1$ and obtain exact expressions of it for $M=1$, $2$ and $3$. For general $M$, we are able to find the exact expression of the spectral density only when all the matrices $X(t)$ for $t=1,\ldots, M$ are square. We also make some observations for general $M$, based admittedly on some weak numerical evidence, which we expect to be correct.

Abstract:
We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. We discuss infinite divisibility in the multiplicative monotonic context as well.

Abstract:
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively the derivation of an analytical expression for the spectra of covariance matrices approximated by free Levy stable random variables and validate it by Monte Carlo simulation.