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Spectral measure of large random Hankel, Markov and Toeplitz matrices. (English) Zbl 1094.15009
In his pioneering work [Ann. Math. (2) 67, 325–327 (1958; Zbl 0085.13203)], E. Wigner proved that the re-scaled spectral measure of symmetric random matrices with independent identically distributed entries converge to the semicircle law. In the same fashion, the authors of the paper under review study the limiting spectral measure of certain random matrices with linear algebraic structure. Namely, they consider Hankel and Toeplitz matrices with independent (besides the obvious constraint of the linear structure) identically distributed entries and unit variance, and they show almost sure, weak convergence to symmetric distributions whose moments they determine fairly explicitly in terms of volumes of solutions of certain systems of linear equations. Markov matrices are also studied, namely random symmetric matrices with independent identically distributed entries and rows with zero sum. In this case the distribution is proven to be the free convolution of the semicircle and the normal law.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60F10 Large deviations 62H10 Multivariate distribution of statistics
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##### References:
 [1] Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677. · Zbl 0949.60077 [2] Biane, P. (1997). On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46 705–718. · Zbl 0904.46045 [3] Bose, A., Chatterjee, S. and Gangopadhyay, S. (2003). Limiting spectral distributions of large dimensional random matrices. J. Indian Statist. Assoc. 41 221–259. [4] Bose, A. and Mitra, J. (2002). Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 111–120. · Zbl 1014.60038 [5] Bożejko, M. and Speicher, R. (1996). Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Math. Z. 222 135–159. · Zbl 0843.60071 [6] 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. · Zbl 1094.15009 [7] Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 155–178 (electronic). · Zbl 1161.15302 [8] Dudley, R. M. (2002). Real Analysis and Probability . Cambridge Univ. Press. · Zbl 1023.60001 [9] Fulton, W. (2000). Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37 209–249 (electronic). · Zbl 0994.15021 [10] Grenander, U. and Szegő, G. (1984). Toeplitz Forms and Their Applications , 2nd ed. Chelsea, New York. · Zbl 0611.47018 [11] Hammond, C. and Miller, S. (2005). Eigenvalue density distribution for real symmetric Toeplitz ensembles. J. Theoret. Probab. 18 537–566. · Zbl 1086.15024 [12] Hiai, F. and Petz, D. (2000). The Semicircle Law , Free Random Variables and Entropy . Amer. Math. Soc., Providence, RI. · Zbl 0955.46037 [13] 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. [14] Lukacs, E. (1970). Characteristic Functions , 2nd ed. Hafner, New York. · Zbl 0201.20404 [15] Mohar, B. (1991). The Laplacian spectrum of graphs. In Graph Theory , Combinatorics , and Applications 2 871–898. Wiley, New York. · Zbl 0840.05059 [16] 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. · Zbl 0794.05005 [17] Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249–286. · Zbl 1039.82020 [18] Sakhanenko, A. I. (1985). Estimates in an invariance principle. In Limit Theorems of Probability Theory . Trudi Inst. Math. 5 27–44, 175. Nauka, Novosibirsk. [19] Sakhanenko, A. I. (1991). On the accuracy of normal approximation in the invariance principle. Siberian Adv. Math. 1 58–91. [20] Sen, A. and Srivastava, M. (1990). Regression Analysis . Springer, New York. · Zbl 0714.62057 [21] Serre, J.-P. (1997). Répartition asymptotique des valeurs propres de l’opérateur de Hecke $$T_p$$. J. Amer. Math. Soc. 10 75–102. JSTOR: · Zbl 0871.11032 [22] Speicher, R. (1997). Free probability theory and non-crossing partitions. Sém. Lothar. Combin. 39 Art. B39c (electronic). · Zbl 0887.46036 [23] Tanny, S. (1973). A probabilistic interpretation of Eulerian numbers. Duke Math. J. 40 717–722. · Zbl 0284.05006 [24] Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325–327. JSTOR: · Zbl 0085.13203
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