The Annals of Probability

Spectral measure of large random Hankel, Markov and Toeplitz matrices

Włodzimierz Bryc, Amir Dembo, and Tiefeng Jiang

Full-text: Open access

Abstract

We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure.

For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γH, γM and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semicircle and normal densities.

For symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m=0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by $\sqrt{2n\log n}$ converges almost surely to 1.

Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 1-38.

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1140191531

Digital Object Identifier
doi:10.1214/009117905000000495

Mathematical Reviews number (MathSciNet)
MR2206341

Zentralblatt MATH identifier
1094.15009

Subjects
Primary: 15A52
Secondary: 60F99: None of the above, but in this section 62H10: Distribution of statistics 60F10: Large deviations

Keywords
Random matrix theory spectral measure free convolution Eulerian numbers

Citation

Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. doi:10.1214/009117905000000495. https://projecteuclid.org/euclid.aop/1140191531


Export citation

References

  • Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611--677.
  • Biane, P. (1997). On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46 705--718.
  • Bose, A., Chatterjee, S. and Gangopadhyay, S. (2003). Limiting spectral distributions of large dimensional random matrices. J. Indian Statist. Assoc. 41 221--259.
  • Bose, A. and Mitra, J. (2002). Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 111--120.
  • Bożejko, M. and Speicher, R. (1996). Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Math. Z. 222 135--159.
  • Bryc, W., Dembo, A. and Jiang, T. (2003). Spectral measure of large random Hankel, Markov and Toeplitz matrices. Expanded version available at http://arxiv.org/abs/math.PR/0307330.
  • Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 155--178 (electronic).
  • Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Univ. Press.
  • Fulton, W. (2000). Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37 209--249 (electronic).
  • Grenander, U. and Szegő, G. (1984). Toeplitz Forms and Their Applications, 2nd ed. Chelsea, New York.
  • Hammond, C. and Miller, S. (2005). Eigenvalue density distribution for real symmetric Toeplitz ensembles. J. Theoret. Probab. 18 537--566.
  • Hiai, F. and Petz, D. (2000). The Semicircle Law, Free Random Variables and Entropy. Amer. Math. Soc., Providence, RI.
  • Lidskiĭ, V. B. (1950). On the characteristic numbers of the sum and product of symmetric matrices. Dokl. Akad. Nauk SSSR (N.S.) 75 769--772.
  • Lukacs, E. (1970). Characteristic Functions, 2nd ed. Hafner, New York.
  • Mohar, B. (1991). The Laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications 2 871--898. Wiley, New York.
  • Nicolas, J.-L. (1992). An integral representation for Eulerian numbers. In Sets, Graphs and Numbers. Colloq. Math. Soc. János Bolyai 60 513--527. North-Holland, Amsterdam.
  • Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249--286.
  • Sakhanenko, A. I. (1985). Estimates in an invariance principle. In Limit Theorems of Probability Theory. Trudi Inst. Math. 5 27--44, 175. Nauka, Novosibirsk.
  • Sakhanenko, A. I. (1991). On the accuracy of normal approximation in the invariance principle. Siberian Adv. Math. 1 58--91.
  • Sen, A. and Srivastava, M. (1990). Regression Analysis. Springer, New York.
  • Serre, J.-P. (1997). Répartition asymptotique des valeurs propres de l'opérateur de Hecke $T_p$. J. Amer. Math. Soc. 10 75--102.
  • Speicher, R. (1997). Free probability theory and non-crossing partitions. Sém. Lothar. Combin. 39 Art. B39c (electronic).
  • Tanny, S. (1973). A probabilistic interpretation of Eulerian numbers. Duke Math. J. 40 717--722.
  • Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325--327.