On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
Benaych-Georges, Florent
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010), p. 644-652 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1214/09-AIHP324
Classification:  46L54,  15A52 
@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},
     volume = {46},
     year = {2010},
     pages = {644-652},
     doi = {10.1214/09-AIHP324},
     mrnumber = {2682261},
     zbl = {1206.46055},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2010__46_3_644_0}
}

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) pp. 644-652. doi : 10.1214/09-AIHP324. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_3_644_0/

[1] T. Banica, S. Belinschi, M. Capitaine and B. Collins. Free Bessel laws. Canad. J. of Math. To appear, 2009. | MR 2779129 | Zbl 1218.46040

[2] S. Belinschi, A note on regularity for free convolutions. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 635-648. | Numdam | MR 2259979 | Zbl 1107.46043

[3] S. Belinschi, F. Benaych-Georges and A. Guionnet. Regularization by free additive convolution, square and rectangular cases. Complex Anal. Oper. Theory. To appear, 2009. | MR 2551632 | Zbl 1187.46055

[4] F. Benaych-Georges. Failure of the Raikov theorem for free random variables. In Séminaire de Probabilités XXXVIII 313-320. Springer, Berlin, 2004. | MR 2126982

[5] F. Benaych-Georges. Infinitely divisible distributions for rectangular free convolution: Classification and matricial interpretation. Probab. Theory Related Fields 139 (2007) 143-189. | MR 2322694 | Zbl 1129.15019

[6] F. Benaych-Georges. Rectangular random matrices, related free entropy and free Fisher's information. J. Oper. Theory. To appear, 2009. | MR 2552088 | Zbl pre05649824

[7] F. Benaych-Georges. Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 (2009) 471-515. | MR 2496440 | Zbl 1171.15022

[8] H. Bercovici and V. Pata. Stable laws and domains of attraction in free probability theory. With an appendix by P. Biane. Ann. Math. 149 (1999) 1023-1060. | MR 1709310 | Zbl 0945.46046

[9] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded supports. Indiana Univ. Math. J. 42 (1993) 733-773. | MR 1254116 | Zbl 0806.46070

[10] H. Bercovici and D. Voiculescu. Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Related Fields 102 (1995) 215-222. | MR 1355057 | Zbl 0831.60036

[11] G. P. Chistyakov and F. Götze. Limit theorems in free probability theory. I. Ann. Probab. 36 (2008) 54-90. | MR 2370598 | Zbl 1157.46037

[12] G. P. Chistyakov and F. Götze. Limit theorems in free probability theory. II. Cent. Eur. J. Math. 6 (2008) 87-117. | MR 2379953 | Zbl 1148.46035

[13] M. Debbah and Ø. Ryan. Multiplicative free convolution and information-plus-noise type matrices. arXiv. (The submitted version of this paper, more focused on applications than on the result we are interested in here, is [14].)

[14] M. Debbah and Ø. Ryan. Free deconvolution for signal processing applications. To appear, 2009.

[15] V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Adisson-Wesley, Cambridge, MA, 1954. | Zbl 0056.36001

[16] F. Hiai and D. Petz. The Semicircle Law, Free Random Variables, and Entropy. Mathematical Surveys and Monographs 77. Amer. Math. Soc., Providence, RI, 2000. | MR 1746976 | Zbl 0955.46037

[17] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge, 2006. | MR 2266879 | Zbl 1133.60003

[18] K. I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 1999. | MR 1739520 | Zbl 0973.60001

[19] D. V. Voiculescu, K. Dykema and A. Nica. Free Random Variables. CRM Monographs Series 1. Amer. Math. Soc., Providence, RI, 1992. | MR 1217253 | Zbl 0795.46049