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Bulk behaviour of some patterned block matrices. (English) Zbl 1351.15021

Summary: We investigate the bulk behaviour of singular values and/or eigenvalues of two types of block random matrices. In the first one, we allow unrestricted structure of order \(m \times p\) with \(n \times n\) blocks and in the second one we allow \(m \times m\) Wigner structure with symmetric \(n \times n\) blocks. Different rows of blocks are assumed to be independent while the blocks within any row satisfy a weak dependence assumption that allows for some repetition of random variables among nearby blocks. In general, \(n\) can be finite or can grow to infinity. Suppose the input random variables are i.i.d. with mean 0 and variance 1 with finite moments of all orders. We prove that under certain conditions, the Marčenko-Pastur result holds in the first model when \(m\to\infty\) and \(\frac{m} {p} \to c \in (0,\infty)\), and the semicircular result holds in the second model when \(m\to\infty\) These in particular generalize the bulk behaviour results of P. Loubaton [“On the almost sure location of singular values of certain Gaussian block-Hankel large random matrices”, to appear in J. Theor. Probab.].

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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