Abstract
We consider the real and complex noncentral Wishart distributions. The moments of these distributions are shown to be expressed as weighted generating functions of graphs associated with the Wishart distributions. We give some bijections between sets of graphs related to moments of the real Wishart distribution and the complex noncentral Wishart distribution. By means of the bijections, we see that calculating these moments of a certain class the real Wishart distribution boils down to calculations for the case of complex Wishart distributions. Nous considérons les lois Wishart non-centrale réel et complexe. Les moments sont décrits comme fonctions génératrices de graphes associées avec les lois Wishart. Nous donnons bijections entre ensembles de graphes relatifs aux moments des lois Wishart non-centrale réel et complexe. Au moyen de la bijection, nous voyons que le calcul des moments d'une certaine classe la loi Wishart réel deviennent le calcul de moments de loi Wishart complexes.
Highlights
First we recall the Wishart distributions which originate from the paper by Wishart [18]
We give some bijections between sets of graphs related to moments of the real Wishart distribution and the complex noncentral Wishart distribution
By means of the bijections, we see that calculating these moments of a certain class the real Wishart distribution boils down to calculations for the case of complex Wishart distributions
Summary
First we recall the Wishart distributions which originate from the paper by Wishart [18]. The moment generating function of the complex Wishart distribution is given as follows: E[etr(ΘW )] = det(I − ΘΣ)−ν e− tr(I−ΘΣ)−1Θ∆, where Θ is a p × p Hermitian parameter matrix. Letac and Massam [9] introduced a method to calculate the moments of the noncentral Wishart distributions. We introduce another formula for the moments of Wishart distribution; in our formula, the moments are described as special values of the weighted generating function of matchings of graphs.
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