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Balanced random and Toeplitz matrices. (English) Zbl 1225.60014
Summary: Except for Toeplitz and Hankel matrices, the common patterned matrices for which the limiting spectral distribution (LSD) is known to exist share a common property – the number of times each random variable appears in the matrix is (more or less) the same across the variables. Thus, it seems natural to ask what happens to the spectrum of the Toeplitz and Hankel matrices when each entry is scaled by the square root of the number of times that entry appears in the matrix instead of the uniform scaling by \(n^{-1/2}\). We show that the LSD of these balanced matrices exist and derive integral formulae for the moments of the limit distribution. Curiously, it is not clear if these moments define a unique distribution

60B20 Random matrices (probabilistic aspects)
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
60G57 Random measures
60B10 Convergence of probability measures
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