Random matrix approximation of spectra of integral operators

Bernoulli

Volumn 6, Issue 1, 2000, Pages 113-167

  Koltchinskii, Vladimir ^a   Giné, Evarist ^b  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Eigenvalues; Heat kernels; Integral operators; Limit theorems; Random matrices

Indexed keywords

EID: 4344709031     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.2307/3318636     Document Type: Article

References (38)

1. Anderson, T.W. (1948) The asymptotic distributions of the roots of certain determinantal equations. J. Roy. Statist. Soc. Ser. B, 10, 132-139.
2. Bai, Z.D. (1993a) Convergence rate of expected spectral distribution of large random matrices. Part I. Wigner matrices. Ann. Probab., 21, 625-648.
3. Bai, Z.D. (1993b) Convergence rate of expected spectral distribution of large random matrices. Part II. Sample covariance matrices. Ann. Probab. 21, 649-673.
4. Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B. (1987) Schrodinger Operators with Applications to Quantum Mechanics and Global Geometry. New York: Springer-Verlag.
5. Davies, E.B. (1989) Heat Kernels and Spectral Tlieory. Cambridge: Cambridge University Press.
6. Dauxois, J., Pousse, A. and Romain, Y. (1982) Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. J. Multivariate Anal, 12, 136-154.
7. Dehling, H. and Mikosch, T. (1994) Random quadratic forms and the bootstrap for [/-statistics. J. Multivariate Anal, 51, 392-413.
8. Dunford, N. and Schwartz, J.T. (1963) Linear Operators. New York: Wiley Interscience.
9. Dyson, F.J. (1962a) Statistical theory of energy levels of complex systems, I. J. Math. Phys., 3, 140-156.
10. Dyson, F.J. (1962b) Statistical theory of energy levels of complex systems, II J. Math. Phys. 3, 157-160.
11. Dyson, F.J. (1962c) Statistical theory of energy levels of complex systems, III. J. Math. Phys. 3, 166-175.
12. Eaton, M.L. and Tyler, D.E. (1991) On Wielandt's inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix. Ann. Statist., 19, 260-271.
13. Geman, S. (1980) A limit theorem for the norm of random matrices. Ann. Probab., 8, 252-261.
14. Gine, E. and Zinn, J. (1992) On Hoffmann-Jorgensen's inequality for {/-processes. In R.M. Dudley, M.G. Hahn and J. Kuelbs (eds), Probability in Danach Spaces 8, pp. 80-91. Boston: Birkhäuser.
15. Gine, E. and Zinn, J. (1994) A remark on convergence in distribution of [/-statistics. Ann. Probab., 22, 117-125.
16. Gine, E. and Zhang, C.-H. (1996) On integrability in the LIL for degenerate [/-statistics. J. Tlieoret. Probab., 9, 385-412.
17. Girko, VL. (1990) Tlieory of Random Determinants. Dordrecht: Kluwer Academic.
18. Gohberg, I. and Krein, M. (1968) Introduction to the Theory of Linear Nonselfadjoint Operators. Providence, RI: American Mathematical Society.
19. Grenander, U. (1963) Probabilities on Algebraic Structures. Stockholm: Almqvist & Wiksell.
20. Hoffman, A.J. and Wielandt, H.W. (1953) The variation of the spectrum of a normal matrix. Duke Math. J., 20, 37-39.
21. Hsu, P.L. (1939) On the distribution of the roots of certain determinantal equations. Ann. Eugenics, 9, 250-258.
22. James, A.T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist., 25, 40-75.
23. Kato, T. (1982) A Short Introduction to Perturbation Theory for Linear Operators. New York: Springer-Verlag.
24. Latala, R. (1998) On the almost sure boundedness of norms of some empirical operators. Statist Probab. Lett., 38, 177-182.
25. Lidskii, V.B. (1950) The proper values of the sum and product of symmetric matrices. Dokl Akad. Nauk SSSR, 75, 769-772.
26. Mehta, M.L. (1991) Random Matrices. New York: Academic Press.
27. Ozawa, S. (1987) Fluctuation of spectra in random media. In K. Ito and N. Ikcda (cds), Probabilistic Methods in Mathematical Physics, pp. 335-361. Boston: Academic Press.
28. Ozawa, S. (1993) Spectra of random media with many randomly distributed obstacles. Osaka J. Math., 30, 1-27.
29. Pastur, L. (1973) Spectra of random selfadjoint operators. Uspekhi Mat. Nank, 28, 3-64. [in Russian].
30. English translation: Russian Math. Surveys, 28(1), 1-67.
31. Silverstein, J.W. (1985) The largest eigenvalue of a large dimensional Wishart matrix. Ann. Probah, 13, 1364-1368.
32. Simon, B. (1979) Functional Integration and Quantum Physics. New York: Academic Press.
33. Simon, B. (1982) Schrödinger semigroups. Bull. Amer. Math. Soc., 7, 447-526.
34. Serfling, R.J. (1980) Approximation Tlieorems in Mathematical Statistics. New York: Wiley.
35. Vardi, Y, Shepp, L.A. and Kaufman, L. (1985) A statistical model for positron emission tomography. J. Amer. Statist. Assoc., 80, 8-22.
36. Voiculescu, D. (1991) Limit laws for random matrices and free products. Invent. Math., 104, 201-220.
37. Wielandt, H. (1967) Topics in the Analytic Tlieory of Matrices (lecture notes prepared by R.R. Meyer). Madison: University of Wisconsin Press.
38. Wigner, E.P. (1955) Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math., 62, 548-564.