## Electronic Communications in Probability

### On the spectral norm of a random Toeplitz matrix

Mark Meckes

#### Abstract

Suppose that $T_n$ is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of $T_n$ is of the order $\sqrt{n \log n}$. The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been studied in the literature.

#### Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 31, 315-325.

Dates
Accepted: 3 October 2007
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465224974

Digital Object Identifier
doi:10.1214/ECP.v12-1313

Mathematical Reviews number (MathSciNet)
MR2342710

Zentralblatt MATH identifier
1130.15018

Subjects
Primary: 15A52
Secondary: 60F99: None of the above, but in this section

Rights

#### Citation

Meckes, Mark. On the spectral norm of a random Toeplitz matrix. Electron. Commun. Probab. 12 (2007), paper no. 31, 315--325. doi:10.1214/ECP.v12-1313. https://projecteuclid.org/euclid.ecp/1465224974

#### References

• Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611–677.
• A. Bose and J. Mitra. Limiting spectral distribution of a special circulant. Statist. Probab. Lett., 60(1):111–120, 2002.
• A. Bose and A. Sen. Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices. Electron. Comm. Probab., 12:29–35, 2007.
• A. Bottcher and B. Silbermann. Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer-Verlag, New York, 1999.
• Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1–38.
• Dudley, R. M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 1967 290–330.
• S. Geman. A limit theorem for the norm of random matrices. Ann. Probab., 8(2):252–261, 1980.
• Halász, G. On a result of Salem and Zygmund concerning random polynomials. Studia Sci. Math. Hungar. 8 (1973), 369–377.
• Hammond, Christopher; Miller, Steven J. Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. 18 (2005), no. 3, 537–566.
• Kashin, B.; Tzafriri, L. Lower estimates for the supremum of some random processes. East J. Approx. 1 (1995), no. 1, 125–139.
• Kashin, B.; Tzafriri, L. Lower estimates for the supremum of some random processes. II. East J. Approx. 1 (1995), no. 3, 373–377.
• Latala, Rafal. Some estimates of norms of random matrices. Proc. Amer. Math. Soc. 133 (2005), no. 5, 1273–1282 (electronic).
• Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120–216, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
• M. Ledoux. The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
• Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces.Isoperimetry and processes.Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
• A. E. Litvak, A. Pajor, M. Rudelson, and N. Tomczak-Jaegermann. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math., 195(2):491–523, 2005.
• Masri, Ibrahim; Tonge, Andrew. Norm estimates for random multilinear Hankel forms. Linear Algebra Appl. 402 (2005), 255–262.
• A. Massey, S. J. Miller, and J. Sinsheimer. Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices. J. Theoret. Probab. To appear. Preprint available at http://arxiv.org/abs/math/0512146
• M. Rudelson. Probabilistic and combinatorial methods in analysis. Lecture notes from an NSF-CBMS Regional Research Conference at Kent State University, 2006.
• Salem, R.; Zygmund, A. Some properties of trigonometric series whose terms have random signs. Acta Math. 91, (1954). 245–301.
• Talagrand, Michel. Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 73–205.
• Talagrand, Michel. Majorizing measures: the generic chaining. Ann. Probab. 24 (1996), no. 3, 1049–1103.
• Talagrand, M. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996), no. 3, 587–600.
• 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 Related Fields 78 (1988), no. 4, 509–521.