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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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