Wishart and anti-Wishart random matrices (original) (raw)
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On a symbolic representation of non-central Wishart random matrices with applications
Journal of Multivariate Analysis, 2014
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.
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Non-Hermitean Wishart random matrices (I)
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A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real-quaternion) stochastic time series representing two 'remote' complex systems. The first paper in a series provides a detailed spectral theory of non-Hermitean Wishart random matrices composed of complex valued entries. The great emphasis is placed on an asymptotic analysis of the mean eigenvalue density for which we derive, among other results, a complex-plane analogue of the Marcenko-Pastur law. A surprising connection with a class of matrix models previously invented in the context of quantum chromodynamics is pointed out.
Random geometric graphs and the spherical Wishart matrix
arXiv (Cornell University), 2021
We consider the random geometric graph on n vertices drawn uniformly from a d-dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of d and n. We show that, when d is larger than n by logarithmic factors, this graph is comparable to the Erdős-Rényi random graph of the same edge density in the inclusion divergence between the graph laws. This divergence functions in certain ways like a relaxation of the total variation distance, but is strong enough to distinguish Erdős-Rényi graphs of different densities with a higher resolution than the total variation distance. To do the analysis, we derive some exact statistics of the spherical Wishart matrix, the Gram matrix of n independent uniformly random d-dimensional spherical vectors. In particular we give expressions for the characteristic function of the spherical Wishart matrix which are well-approximated using steepest descent.