On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Studia Mathematica

Volumn 195, Issue 1, 2009, Pages 11-29

  Pajor, A ^a   Pastur, L ^b  

^a UNIVERSITÉ GUSTAVE EIFFEL   (France)

^b B VERKIN INSTITUTE FOR LOW TEMPERATURE PHYSICS AND ENGINEERING   (Ukraine)

Author keywords

Asymptotic theory of convex bodies; Eigenvalue distribution; Log concave measures; Random matrices

Indexed keywords

EID: 72549087386     PISSN: 00393223     EISSN: None     Source Type: Journal    
DOI: 10.4064/sm195-1-2     Document Type: Article

References (28)

1. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993.
2. G. Aubrun, Random points in the unit ball of ℓpn, Positivity 10 (2006), 755-759.
3. -, Sampling convex bodies: a random matrix approach, Proc. Amer. Math. Soc. 135 (2007), 1293-1303.
4. Z. D. Bai, Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist. Sinica 9 (1999), 611-677.
5. C. Borell, Convex measures on locally convex spaces, Ark. Mat. 12 (1974), 239-252.
6. -, Convex set functions in d-space, Period. Math. Hungar. 6 (1975), 111-136.
7. J. Bourgain, Random points in isotropic convex sets, in: Convex Geometric Analysis (Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ. 34, Cambridge Univ. Press, Cambridge, 1999, 53-58.
8. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol.1, Interscience, New York, 1953.
9. Ju. S. Davidovič, B. I. Korenbljum, and B. I. Hacet, A certain property of logarithmically concave functions, Dokl. Akad. Nauk SSSR 185 (1969), 1215-1218 (in Russian).
10. English transl.: Soviet Math. Dokl. 10 (1969), 447-480.
11. K. R. Davidson and S. Szarek, Local operator theory, random matrices and Banach spaces, in: Handbook of the Geometry of Banach Spaces, Vol.1, W. B. Johnson and J. Lindenstrauss (eds.), Elsevier, 2001, 317-366.
12. B. Efron and C. Stein, The jackknife estimate of variance, Ann. Statist. 9 (1981), 586-596.
13. B. Fleury, O. Guédon and G. Paouris, A stability result for mean width of Lp-centroid bodies, Adv. Math. 214 (2007), 865-877.
14. J. S. Geronimo and T. P. Hill, Necessary and sufficient condition that the limit of Stieltjes transforms is the Stieltjes transform, J. Approx. Theory 121 (2003), 54-60.
15. A. Giannopoulos and V. Milman, Concentration property on probability spaces, Adv. Math. 156 (2000), 77-106.
16. A. Giannopoulos and A. Tsolomitis, Volume radius of a random polytope in a convex body, Math. Proc. Cambridge Philos. Soc. 134 (2003), 13-21.
17. V. L. Girko, Theory of Stochastic Canonical Equations, Vols. I, II, Kluwer, Dordrecht, 2001.
18. B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 (2007), 284-310.
19. -, A central limit theorem for convex sets, Invent. Math. 168 (2007), 91-131.
20. A. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005), 491-523.
21. V. A. Marchenko and L. A. Pastur, The eigenvalue distribution in some sets of random matrices, Math. USSR-Sb. 1 (1967), 457-483.
22. V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, New York, 1986.
23. R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982.
24. G. Paouris, Concentration of mass of isotropic convex bodies, C. R. Math. Acad. Sci. Paris 342 (2006), 179-182.
25. L. Pastur, A simple approach to the global regime of the random matrix theory, in: Mathematical Results in Statistical Mechanics, S. Miracle-Sole et al. (eds.), World Sci., Singapore, 1999, 429-454.
26. A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301-316.
27. J. W. Silverstein and Z. D. Bai, On the limiting empirical distribution of eigenvalues of a class of large dimensional random matrices, J. Approx. Theory 54 (1995), 175-192.
28. S. Szarek, Convexity, complexity, and high dimensions, in: Proc. Int. Congress of Mathematicians (Madrid, 2006), Vol.II, Eur. Math. Soc, 2006, 1599-1621.