On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions

Florent Benaych-Georges

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 644-652
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

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 46.3 (2010): 644-652. <http://eudml.org/doc/240980>.

@article{Benaych2010,
abstract = {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.},
author = {Benaych-Georges, Florent},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {free probability; random matrices; free convolution; infinitely divisible laws; Marchenko–Pastur law; Marchenko-Pastur law},
language = {eng},
number = {3},
pages = {644-652},
publisher = {Gauthier-Villars},
title = {On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions},
url = {http://eudml.org/doc/240980},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Benaych-Georges, Florent
TI - On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 644
EP - 652
AB - 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.
LA - eng
KW - free probability; random matrices; free convolution; infinitely divisible laws; Marchenko–Pastur law; Marchenko-Pastur law
UR - http://eudml.org/doc/240980
ER -

References

top
  1. [1] T. Banica, S. Belinschi, M. Capitaine and B. Collins. Free Bessel laws. Canad. J. of Math. To appear, 2009. Zbl1218.46040MR2779129
  2. [2] S. Belinschi, A note on regularity for free convolutions. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 635–648. Zbl1107.46043MR2259979
  3. [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. Zbl1187.46055MR2551632
  4. [4] F. Benaych-Georges. Failure of the Raikov theorem for free random variables. In Séminaire de Probabilités XXXVIII 313–320. Springer, Berlin, 2004. Zbl1076.46050MR2126982
  5. [5] F. Benaych-Georges. Infinitely divisible distributions for rectangular free convolution: Classification and matricial interpretation. Probab. Theory Related Fields 139 (2007) 143–189. Zbl1129.15019MR2322694
  6. [6] F. Benaych-Georges. Rectangular random matrices, related free entropy and free Fisher’s information. J. Oper. Theory. To appear, 2009. Zbl1224.46121MR2552088
  7. [7] F. Benaych-Georges. Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 (2009) 471–515. Zbl1171.15022MR2496440
  8. [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. Zbl0945.46046MR1709310
  9. [9] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded supports. Indiana Univ. Math. J. 42 (1993) 733–773. Zbl0806.46070MR1254116
  10. [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. Zbl0831.60036MR1355057
  11. [11] G. P. Chistyakov and F. Götze. Limit theorems in free probability theory. I. Ann. Probab. 36 (2008) 54–90. Zbl1157.46037MR2370598
  12. [12] G. P. Chistyakov and F. Götze. Limit theorems in free probability theory. II. Cent. Eur. J. Math. 6 (2008) 87–117. Zbl1148.46035MR2379953
  13. [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. [14] M. Debbah and Ø. Ryan. Free deconvolution for signal processing applications. To appear, 2009. 
  15. [15] V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Adisson-Wesley, Cambridge, MA, 1954. Zbl0056.36001
  16. [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. Zbl0955.46037MR1746976
  17. [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. Zbl1133.60003MR2266879
  18. [18] K. I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 1999. Zbl0973.60001MR1739520
  19. [19] D. V. Voiculescu, K. Dykema and A. Nica. Free Random Variables. CRM Monographs Series 1. Amer. Math. Soc., Providence, RI, 1992. Zbl0795.46049MR1217253

NotesEmbed ?

top

You must be logged in to post comments.