Bulk behaviour of some patterned block matrices

Debapratim Banerjee () and Arup Bose
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Debapratim Banerjee: Indian Statistical Institute

Indian Journal of Pure and Applied Mathematics, 2016, vol. 47, issue 2, 273-289

Abstract: Abstract 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 × p with n × n blocks and in the second one we allow m × m Wigner structure with symmetric n × 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 → ∞ and $$\tfrac{m} {p} \to c \in (0,\infty )$$ , and the semicircular result holds in the second model when m → ∞. These in particular generalize the bulk behaviour results of Loubaton [10].

Keywords: Block matrix; Hankel matrix; Toeplitz matrix; symmetric circulant matrix; Wigner matrix; limit spectral distribution; semi-circle law; Marchenko-Pastur law; Carleman’s condition (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s13226-016-0187-2

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