The limiting spectra of Girko's block-matrix (Q2471124)

For \(n,k\geqslant 1\) the author considers the double array of random block-matrices \(\left \{ B_{n,k}\right \} \) whose terms are given by \(B_{n,k}=I_{k}\oplus A_{n}+W_{k}\oplus B_{n},\) where for \(n\geqslant 1\), the matrices \(A_{n},B_{n}\) and \(Wn\) are Hermitian random matrices of order \(n\), and satisfy the following hypotheses: (i) There exists a compactly supported probability measure \(\mu _{\omega }\) such that \(\mu W_{n}\overset{m}{\rightarrow }\mu _{\omega }\) as \(n\rightarrow \infty \) a.s. (ii) For real \(t\), there exists probability measures \(\Psi (t,\cdot)\) such that \[ \mu _{A_{n}}+tB_{n}\overset{m}{\rightarrow }\Psi (t,\cdot) \text{ as }n\rightarrow \infty \text{ a.s.} \] Under these conditions \[ \lim_{n\rightarrow \infty}\lim_{k\rightarrow \infty} \mu _{B_{n,k}}\overset{\infty }{=} \lim_{k\rightarrow \infty}\lim_{n\rightarrow \infty} \mu _{B_{n,k}}\overset{\infty }{=}\nu \text{ a.s.}, \] where the probability measure \(\nu \) is defined as \[ \nu (dx) =\int_{R}\Psi (t;dx) \mu _{\omega}(dt). \] The main tool of the proof is the method of moments. As applications of the theorem, several propositions are proved when the blocks are made of the known ensembles like the Gaussian unitary ensemble and the Wishart random matrix. The free probability theory is used to prove these propositions.