On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 644-652.

Dans cet article, on prouve un résultat reliant les versions carré et rectangulaire de la R-transformée, qui a pour conséquence une relation surprenante entre les versions carré et rectangulaire de la convolution libre additive, impliquant la loi de Marchenko-Pastur. On donne des conséquences de ce résultat portant sur les matrices aléatoires, sur l'infinie divisibilité et sur l'arithmétique des versions carré des convolutions additives et multiplicatives.

In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.

DOI : https://doi.org/10.1214/09-AIHP324
Classification : 46L54,  15A52
Mots clés : free probability, random matrices, free convolution, infinitely divisible laws, Marchenko-Pastur law
@article{AIHPB_2010__46_3_644_0,
     author = {Benaych-Georges, Florent},
     title = {On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {644--652},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {3},
     year = {2010},
     doi = {10.1214/09-AIHP324},
     zbl = {1206.46055},
     mrnumber = {2682261},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/09-AIHP324/}
}
Benaych-Georges, Florent. On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 644-652. doi : 10.1214/09-AIHP324. http://archive.numdam.org/articles/10.1214/09-AIHP324/

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