Bulk behaviour of some patterned block matrices

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 mp→c∈(0,∞) , and the semicircular result holds in the second model when m → ∞. These in particular generalize the bulk behaviour results of Loubaton [10].