Afrika Statistika

R-transform associated with asymptotic negative spectral moments of Jacobi ensemble

Jolanta Pielaszkiewicz

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We derive an explicit formula for the R-transform of inverse Jacobi matrix $I+W_1^{-1}W_2$, where $W_1,W_2\sim\mathcal{W}_p(I,n_i)$, $i=1,2$ are independent and $I$ is $p\times p$ dimensional identity matrix using property of asymptotic freeness of Wishart and deterministic matrices. Procedure can be extended to other sets of the asymptotically free independent matrices. Calculations are illustrated with some simulations on fixed size matrices.


Nous dérivons une formule explicite pour une transformée en R de $I+W_1^{-1}W_2$, où $W_1,W_2\sim\mathcal{W}_p(I,n_i)$, $i=1,2$, et $I$ correspondent à une matrice identité de dimension $p\times p$ en utilisant les propriétés de la loi asymptotique de Wishart et des matrices déterministes. La procédure peut être étendue à d'autres ensembles de matrices indépendantes asymptotiquement libres. Les calculs sont illustrés avec des simulations sur les matrices de dimension fixe.

Article information

Afr. Stat., Volume 13, Number 1 (2018), 1531-1538.

First available in Project Euclid: 17 May 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 46L54: Free probability and free operator algebras

Jacobi ensemble R-transform S-transform Negative spectral moments Spectral moments Wishart matrix Marčenko-Pastur law asymptotical freeness


Pielaszkiewicz, Jolanta. R-transform associated with asymptotic negative spectral moments of Jacobi ensemble. Afr. Stat. 13 (2018), no. 1, 1531--1538. doi:10.16929/as/1531.118.

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