Enumerative combinatorics
SIAM Journal on Mathematical Analysis - Special issue: the articles in this issue are dedicated to Richard Askey and Frank Olver
Expectations of useful complex Wishart forms
Multidimensional Systems and Signal Processing
MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions
Advances in Applied Mathematics
Approximations to the distribution of the sample correlation matrix
Journal of Multivariate Analysis
An umbral setting for cumulants and factorial moments
European Journal of Combinatorics
The noncentral Wishart as an exponential family, and its moments
Journal of Multivariate Analysis
Cumulants and convolutions via Abel polynomials
European Journal of Combinatorics
Note(s): Moments and cumulants for the complex Wishart
Journal of Multivariate Analysis
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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 traces of its central component and of a formal variable 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.