Canonical Equation K 32 for Normalized Spectral Functions of Random Gram Matrices with Identically Distributed Independent Blocks. Block Matrix Density

Abstract

The Chapter extends the “One quarter Law” \( {(2\pi )^{ - 1}}\sqrt {(4 - x)} \;{x^{ - 1/2}},\;{\rm{0}}\;{\rm{ < }}\;x {\rm{ < }}\;{\rm{4}} \) to Gram random matrices with independent random blocks obeying a Lindeberg-type condition and allowing arbitrary dependence of entries within each block. It is proved that Stieltjes transform of the individual limiting spectral function satisfies a matrix canonical equation which generalizes the Bronk-Marchenko-Pastur density. We make the same assumptions concerning random matrices: we only change all entries of random matrices by some blocks.