Concentration of the spectral measure for large random matrices with stable entries

Electronic Journal of Probability

Volumn 13, Issue , 2008, Pages 107-134

  Houdré, Christian ^a   Xu, Hua ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Concentration; Infinitely divisibility; Random Matrices; Spectral Measure; Stable Vector

Indexed keywords

EID: 39449133672     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v13-482     Document Type: Article

References (35)

1. N. Alon, M. Krivelevich, V. Vu, On the concentration of eigenvalues of random symmetric matrices. Isr. J. Math., 131, (2002), 259-267.
2. G.W. Anderson, A. Guionnet, O. Zeitouni, Lecture notes on random matrices. SAMSI. September, (2006).
3. Z.D. Bai, Circular law. Ann. Probab., 25, (1997), 494-529.
4. Z.D. Bai, J.W. Silverstein, Spectral analysis of large dimensional random matrices. Science Press, Beijing, (2006).
5. J.-C. Breton, C. Houdre, On finite range stable-type concentration. To appear Theory Probab. Appl. (2007).
6. J.-C. Breton, C. Houdre, N. Privault, Dimension free and infinite variance tail estimates on Poisson space. Acta Appli. Math., 95, (2007), 151-203.
7. R.M. Dudley, Real analysis and probability. Cambridge University Press, (2002).
8. V.L. Girko, Spectral theory of random matrices. Moscow, Nauka, (1988).
9. F. Götze, A. Tikhomirov, On the circular law. Preprint. Avalaible at Math arXiv 0702386, (2007).
10. A. Guionnet, O. Zeitouni, Concentration of spectral measure for large matrices. Electron. Comm. Probab. 5 (2000), 119-136.
11. R.A. Horn, C. Johnson, Topics in matrix analysis. Cambridge Univ. Press, (1991).
12. C. Houdre, Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 (2002), 1223-1237.
13. C. Houdre, P. Marchal, On the concentration of measure phenomenon for stable and related random vectors. Ann. Probab. 32 (2004), 1496-1508.
14. C. Houdre, P. Marchal, Median, concentration and fluctuations for Levy processes. Stochastic Processes and their Applications (2007).
15. C. Houdre, P. Marchal, P. Reynaud-Bouret, Concentration for norms of infinitely divisible vectors with independent components. Bernoulli (2008).
16. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices. Duke math. J. 91 (1998), 151-204.
17. I.M. Johnstone, On the distribution of the largest principal component. Ann. Statist., 29, (2001), 295-327.
18. I.M. Johnstone, High dimensional statistical inference and random matrices. International Congress of Mathematicians. Vol. I, 307-333, Eur. Math. Soc., Zürich, (2007)
19. M. Ledoux, The concentration of measure phenomenon. Math. Surveys Monogr., 89 American Mathematical Society, Providence, RI, (2001).
20. M. Ledoux, Deviation inequalities on largest eigenvalues. Geometric Aspects of Functional Analysis, Israel Seminar 2004-2005, Lecture Notes in Math. 1910, 167-219. Springer (2007).
21. P. Marchal, Measure concentration for stable laws with index close to 2. Electron. Comm. Probab. 10 (2005), 29-35.
22. M.B. Marcus, J. Rosinski, Lorms of infinitely divisible random vectors and certain stochastic integrals. Electron. Comm. Probab. 6 (2001), 15-29.
23. V.A. Marcenko, L.A. Pastur. Distributions of eigenvalues of some sets of random matrices. Math. USSR-Sb., 1 (1967), 507-536.
24. M.L. Mehta, Random matrices, 2nd ed. Academic Press, San Diego, (1991).
25. L.A. Pastur, On the spectrum of random matrices. Teor. Mat. Fiz. 10 (1972), 102-112.
26. B. Simon, Trace ideals and their applications. Cambridge University Press, (1979).
27. K-I. Sato, Levy processes and infinitely divisible distributions. Cambridge University Press, Cambridge, (1999).
28. A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999), 697-733.
29. A. Soshnikov, A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Stat. Phys., 108, Nos 5/6, (2002), 1033-1056.
30. A. Soshnikov, Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Elec. Commun. Probab. 9 (2004), 82-91.
31. A. Soshnikov, Poisson statistics for the largest eigenvalues in random matrix ensembles. Mathematical physics of quantum mechanics, 351-364, Lecture Notes in Phys., 690, Springer, Berlin, 2006.
32. A. Soshnikov, Y. V. Fyodorov, On the largest singular values of random matrices with independent Cauchy entries. J. Math. Phys. 46 (2005), 033302.
33. C. Tracy, H. Widom, Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 (1994), 151-174.
34. C. Tracy, H. Widom, On orthorgonal and symplectic random matrix ensembles. Comm. Math. Phys. 177 (1996), 724-754.
35. E.P. Wigner, On the distribution of the roots of certain symmetric matrices. Ann. Of Math. (2) 67 (1958), 325-327.