Abstract
Let \({{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}\) be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices \({{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}\) , possibly random but independent of X N , for which the operator norm of \({P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}\) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y N and of the polynomials P, we get for a large class of matrices the “no eigenvalues outside a neighborhood of the limiting spectrum” phenomena. We give examples of diagonal matrices Y N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.
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Anderson G.W., Guionnet A., Zeitouni O.: An introduction to random matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)
Arnold L.: On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20, 262–268 (1967)
Bai, Z.D.: Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sin. 9(3), 611–677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author (1999)
Bai Z.D., Silverstein J.W., Yin Y.Q.: A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivar. Anal. 26(2), 166–168 (1988)
Bai Z.D., Silverstein J.W.: No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26(1), 316–345 (1998)
Bai Z.D., Yin Y.Q.: Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16(4), 1729–1741 (1988)
Bai Z.D., Yin Y.Q., Krishnaiah P.R.: On limiting spectral distribution of product of two random matrices when the underlying distribution is isotropic. J. Multivar. Anal. 19(1), 189–200 (1986)
Brown N.P., Ozawa N.: C*-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence (2008)
Capitaine M., Casalis M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices. Indiana Univ. Math. J. 53(2), 397–431 (2004)
Capitaine M., Donati-Martin C.: Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56(2), 767–803 (2007)
Chen L.H.Y.: An inequality for the multivariate normal distribution. J. Multivar. Anal. 12(2), 306–315 (1982)
Conway J.B.: A course in operator theory. Graduate Studies in Mathematics, vol. 21. American Mathematical Society, Providence (2000)
Dykema K.: On certain free product factors via an extended matrix model. J. Funct. Anal. 112(1), 31–60 (1993)
Füredi Z., Komlós J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)
Geman S.: A limit theorem for the norm of random matrices. Ann. Probab. 8(2), 252–261 (1980)
Grenander U., Silverstein J.W.: Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32(2), 499–519 (1977)
Guionnet, A.: Large random matrices: lectures on macroscopic asymptotics. Lecture Notes in Mathematics, vol. 1957. Springer, Berlin, 2009. Lectures from the 36th Probability Summer School held in Saint-Flour (2006)
Guionnet, A., Krishnapur, M., Zeitouni, O.: The single-ring theorem. arXiv:0909.2214v1 (preprint). http://www.arxiv4.library.cornell.edu/abs/0909.2214
Haagerup U., Thorbjørnsen S.: A new application of random matrices: \({{\rm Ext}(C^*_{\rm red}(\mathbb F_2))}\) is not a group. Ann. Math. (2) 162(2), 711–775 (2005)
Hamburger H.: Über eine Erweiterung des Stieltjesschen Momentenproblems. Math. Ann. 82(3–4), 168–187 (1921)
Hiai F., Petz D.: Asymptotic freeness almost everywhere for random matrices. Acta Sci. Math. (Szeged) 66(3–4), 809–834 (2000)
Jonsson D.: Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivar. Anal. 12(1), 1–38 (1982)
Jonsson, D.: On the largest eigenvalue of a sample covariance matrix. In: Multivariate Analysis VI (Pittsburgh, PA, 1983), pp. 327–333. North-Holland, Amsterdam (1985)
Juhász, F.: On the spectrum of a random graph. In: Algebraic Methods in Graph Theory, vols. I, II (Szeged, 1978), Colloq. Math. Soc. János Bolyai, vol. 25, pp. 313–316. North-Holland, Amsterdam (1981)
Larsson E.G., Stoica P.: Space-time Block Coding for Wireless Communications. Cambridge University Press, Cambridge (2003)
Lehner F.: Computing norms of free operators with matrix coefficients. Am. J. Math. 121(3), 453–486 (1999)
Marčenko V.A., Pastur L.A.: Distribution of eigenvalues in certain sets of random matrices. Math. Sb. (N.S.) 72(114), 507–536 (1967)
Nica, A., Speicher, R.: Lectures on the combinatorics of free probability. In: London Mathematical Society Lecture Note Series, vol 335. Cambridge University Press, Cambridge (2006)
Paul D., Silverstein J.W.: No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix. J. Multivar. Anal. 100(1), 37–57 (2009)
Pimsner, M.V.: A class of C*-algebras generalizing both Cuntz–Krieger algebras and crossed products by Z. In: Free Probability Theory (Waterloo, ON, 1995). Fields Institute Communications, vol. 12, pp. 189–212. American Mathematical Society, Providence (1997)
Schultz H.: Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Relat. Fields 131(2), 261–309 (2005)
Shlyakhtenko D.: Some applications of freeness with amalgamation. J. Reine Angew. Math. 500, 191–212 (1998)
Silverstein J.W.: The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab. 13(4), 1364–1368 (1985)
Silverstein J.W.: On the weak limit of the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivar. Anal. 30(2), 307–311 (1989)
Thorbjørnsen S.: Mixed moments of Voiculescu’s Gaussian random matrices. J. Funct. Anal. 176(2), 213–246 (2000)
Tulino, A.M., Verdú, S.: Random matrices and wireless communications. Fundations and Trends in Communications and Information Theory, vol. 1. Now Publishers Inc. (2004)
Voiculescu D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)
Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. Astérisque (232):243–275, Recent advances in operator algebras (Orléans, 1992) (1995)
Voiculescu D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Notices 1, 41–63 (1998)
Wachter K.W.: The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6(1), 1–18 (1978)
Wigner E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. (2) 67, 325–327 (1958)
Yin Y.Q.: Limiting spectral distribution for a class of random matrices. J. Multivar. Anal 20(1), 50–68 (1986)
Yin Y.Q., Bai Z.D., Krishnaiah P.R.: On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Relat. Fields 78(4), 509–521 (1988)
Yin Y.Q., Krishnaiah P.R.: A limit theorem for the eigenvalues of product of two random matrices. J. Multivar. Anal. 13(4), 489–507 (1983)
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With an appendix by Dimitri Shlyakhtenko.
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Male, C. The norm of polynomials in large random and deterministic matrices. Probab. Theory Relat. Fields 154, 477–532 (2012). https://doi.org/10.1007/s00440-011-0375-2
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DOI: https://doi.org/10.1007/s00440-011-0375-2